WHY DO MATHEMATICIANS ACCEPT THIS NONSENSE? IT IS THE ROOT OF

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>continuing a sequence indefinitely is nonsense
>but arbitrarily halting the progression of a sequence is also nonsense
hmmmmmm

What's wrong with it?

>>8820691
INFINITY DOESN'T EVEN EXIST. SHOW ME A SET OF INFINITE APPLES

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>>8820699
Let {A} be the set of all apples that will grow under the following conditions:
1. Apples never go extinct
2. Apple farmers never go extinct

>>8820699
Infinite apples don't exist, so the set of infinite apples is the empty set which can be presented as a set of zero apples.

>>8820688
THE AXIOM OF INFINITY DOESN'T SAY SHIT ABOUT SEQUENCES. IT STATES THAT THERE IS A SET WITH INFINITE ELEMENTS WHICH DOESN'T EXIST IN THE REAL WORLD ONLY IN MATHEMATICIANS IMAGINATION.

>>8820702
>1. Apples never go extinct
>2. Apple farmers never go extinct

NOT REAL WORLD, THAT'S FANTASY. AND ANYWAY WITH THAT CONDITIONS, AT ANY GIVEN TIME THERE WOULD BE SOME FINITE NUMBER OF APPLES IN THE SET, NOT INFINITY

>>8820703
>the set of infinite apples is the empty set

I BET YOU ARE ONE OF THE GUYS WHO ALSO CLAIMS THAT THE SUM OF ALL NATURAL NUMBERS IS NOT ONLY A NEGATIVE NUMBER BUT ALSO A FUCKING FRACTION

>>8820705
There are no perfect isosceles triangles in the real world, either. I guess we should throw away trigonometry and just bang rocks together and make grunting noises for the rest of eternity.

>>8820685
Hello OP, I am a representative from the Vatican. Unfortunately, your thread is heretical and against His Holy Church. Infinity is not to be questioned, such is His will.
Your location has been logged. Please cease the creation of all threads and posts regarding infinity or we will dispatch a field team to remove you.

>>8820685
Go away Norman

>>8820721

SEE THIS IS WHAT HAPPENS WHEN YOU TRY TO EXPOSE MATHEMATICIANS LIES, THEIR FRIENDS TRY TO SILENCE YOU

>>8820688
Lets take a round object.

We measure its diameter to be r

we then begin measuring around the circumference, marking off distances 3. We cut out the sector between 3 and 0. Did we cut into some extra demension, or is there a cut at a fixed point?

pi is rational.
QED

>>8820699
Neither do imaginary apples, nor negative apples. Shall we toss those as well?

>>8820762
YES

Axiom of infinity is not the root of set theory's problems; the real culprit is the axiom of foundation.

Instead of outright banning all infinite descending membership chains they should have assumed instead that every non-empty set contains at least one finite descending chain.

>>8820736
What's the area of an apple?

>>8820705
It doesn't exist in THIS world

You think I'm joking but I'm not

>>8820818
The unmemed memer is infinity

>>8820699
> Implying apples exist.

>>8820818
>It doesn't exist in THIS world

THIS IS THE TYPE OF ARGUMENTS MATHEMATICIANS USE TO JUSTIFY THEIR NONSENSE.

>MUH OTHER WORLDS
>MUH TIME TRAVEL
>MUH APPLES THAT NEVER GO EXTINCT

Point two mirrors at each other. What is the resulting image if not real world infinity? Yes it's not perfect like no real world sphere is perfect, but the underlying 'mechanism' that is responsible for this effect is very much real.

>>8820850
BULLSHIT THERE ARE NO 100% REFLECTIVE MIRRORS AND SO THE LIGHT CAN'T KEEP BOUNCING INFINITELY BETWEEN THEM

NEXT

>>8820838
This is next.
Come at me bro.

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>>8820838
Asking the real questions here.

>>8820740
This only proves pi is real, not rational. Only way to express this is (pi-3)/pi, not as a proper fraction

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>>8820717
>I BET
there is no wagering at 4chan, Grandpa pls

>>8820859
You didn't even try to understand. Never mind

>>8820713
>AT ANY GIVEN TIME THERE WOULD BE SOME FINITE NUMBER OF APPLES IN THE SET, NOT INFINITY

what is the most amount of apples there will ever be?

>>8820685
The universe is infinite. The event horizon of a black hole is a perfect circle. Get bent loser.

>>8820958
>>8820966
you're idiots

>>8820838
Seriosu they're just an emergent phenomenon of interactions between fields

>>8820705
All maths aren't real.

>>8820840
time travel?
other worlds?
he is talking about plato's world of forms, the world of abstract objects you brainlet

>>8820685
>WHY DO MATHEMATICIANS ACCEPT THIS NONSENSE?
Because it creates very nice theories and fits the way we view the world very well.
We dont see space as something disjointed and made up of finite parts even if it is completely discrete.

Unless finitists provide alternatives to stuff like DEs and Integrals which lead to equally nice results and are as applicable to model reality it will stay irrelevant.

>>8820699
Numbers don't even exist, my dude. Neither do sets, functions, or even logic. They're just things we make up.

>>8820685
>>8820699
>>8820705
>>8820713
>>8820717
>>8820778
>>8820736
>>8820840
>>8820859
take your meds and get some rest Grandpa, pls

>>8820685
They accept it because it's useful. Real Numbers makes calc really easy, and calc is really useful.

>>8820685
Sure, there probably isn't an infinite amount of anything in the universe, unless the universe in infinite, but surely you'd want items is the universe to belong to the same set? And you simply can't have an upper bound on that set, so in effect there infinite sets that objects belong to.

>>8820685
>implying math has anything to do with the real world
lol

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>>8820699
infinity exists, apples don't

>>8820699
The set of all apples that are, have been, and possibly may be.

>>8821531
The only thing you can say truly exists is what you can see right in front of you. And even that can't be 100% proven that its real. How can you be so sure you're not currently trapped in an insane asylum just imagining being free from the confides of your padded room?

>>8821629
>How can you be so sure you're not currently trapped in an insane asylum just imagining being free from the confides of your padded room?
I can because only rationalists discriminate between imagination and ''reality'' and because claiming that something is imagined is already claiming that that it is not real and that there is a procedure to see how unreal it is

>>8820685
The set of all seconds from the beginning of the universe to it's end.

>>8820720

The appearance of material objects has no correlation with the collective conscious knowledge of any and all numerological reductions of said objects.

>tfw this kind of discussion is as close as most stemlords will ever get to looking behind the veil

>>8820685
It gives us interesting mathematics. Honestly, what better way is there for picking axioms?

>>8820963
UNDEFINED. BECAUSE IT IS AN IMPOSSIBLE SITUATION THATQUESTION MAKES NO SENSE

>>8820966
>The universe is infinite.
NOT PROVEN. ALSO IF WE LIVED IN SOME GIANT PLANET IT WOULD LOOK LIKE IT NEVER ENDS BUT THAT DOESN'T MEAN IS INFINITE

>>8821344
>he is talking about plato's world of forms, the world of abstract objects
NOT REAL WORLD

>>8821372
BUT NATURAL NUMBERS REPRESENT ACTUAL THINGS OF THE REAL WORLD, SO THEY'RE ACTUAL NUMBERS, THE REST IS IMAGINARY CRAP

>>8821609
>>8821685
VERY LARGE NUMBER != INFINITY

>>8820685
Good to see you again, CAPfag. Still fighting the good fight I see.

>>8821732
SERIOUS MATHEMATICIANS SHOULD START A NEW SYSTEM WITHOUT THE INFINITY AXIOM, THERE IS A REASON WHY MOST PEOPLE HAVE A HARD TIME GETTING THE CONCEPT OF INFINITY IN MATHEMATICS. BECAUSE IT IS MADE UP CRAP

>>8821886
Do you agree with the concepts of 1, 2, 3, 28148218, etc? Do you agree that if x is a number then we have an algorithm to generate a representation for x+1 and know that it is a number as well? Do you agree that it thus makes sense to talk about the set of all such numbers? This is the axiom of infinity. You could literally replace it by "the natural numbers are a set" and you get the same theory.

>>8821890
>Do you agree with the concepts of 1, 2, 3, 28148218, etc?

YES, BUT ACTUAL NUMBERS REPRESENT REAL WORLD THINGS

>Do you agree that if x is a number then we have an algorithm to generate a representation for x+1 and know that it is a number as well?

YES BUT AT SOME POINT X+1 STOPS BEING AN ACTUAL NUMBER AND BECOMES IMAGINARY CRAP SINCE IT DOESN'T REPRESENT ANYTHING IN THE REAL WORLD.THERE IS NO REAL WORLD INFINITY

>>8820685
You can let it fall, no problem. But then you drop out of the level of which ZFC is part of in the godel hierarchy (you end up in the same level as basic number theory, if I'm not wrong. So, a really weak system).

Math is an abstract world, as other anons already tried to explain to you, nothing (math concepts) exists in "reality" (although, concerning infinity, one can argue: If I can think of infinity, and my thoughts are "real", then infinity exists in reality). Natural numbers don't exist in reality: Assume there is no infinity in the real world. Then the set of all things in the universe is of finite cardinality N, but what is N+1 then? Does not exist in reality but is a natural number.

>>8821912
>YES
no you don't agree you imbecile. you don't agree that if X is a number then X+1 is a number. that's literally the axiom of inifinity

Wildburger pls go

>>8820705
Math isn't concerned with what is real. Only with logical structures.

>>8821759
Natural numbers don't represent actual things. There're just symbols used in the placeholder of counting. You can show me 3 of something but you can't show me 3 itself.

>>8821759
>implying time flow will ever end

>>8821959
TOO LARGE NUMBERS DON'T EXIST

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>>8822427
>there exists some arbitrary point such that x+1 isn't a number
But y tho

>>8822435
BECAUSE AT SOME POINT THE ``NUMBERS'' BECOME IMAGINARY CRAP, IT'S LIKE SAYING SOMEONE HAS -5 BROTHERS

>>8822447
>at some point
Which point? And that's a really shit analogy

where do you find numbers in the real world ?

>>8822455
obviously you can't say at which point, because the point doesn't exist

>>8822500
I wonder what the consequences of such an axiom would be

>>8822528
Math would obviously be totally broken. You have to drop the union-axiom, power-set-axiom, pairing-axiom etc.,.... And there is nothing interesting going on there.

>>/lit/
Stay in your containment board, Rei

>>8822447
Do you believe in the existence of pairs? I. E. Can I take two things, say two stones, and consider them together as one, as a pair of stones.

Lewis Carrol pls leave

I'm positive i've seem this exact same thread before.

are we in the Berenstein universe?

Memes aside, I started only recently to have an axiomatic study of set theory.
When I got to the infinity axiom I was.... disapointed. I have no problems with the concept of infinity, but I was hoping to be proved in a more elegant way than just taking an axiom like this, perhaps based on some more simple intuitive axioms.

>>8823720
> I was hoping to be proved in a more elegant way than just taking an axiom like this

If you drop it, you can't proof it from ZFC: (ZFC-INF) =/=> (ZFC).
If your axioms only allow sets with finite cardinality, you can never get to a set with infinite cardinality.

>>8821319
How can apples be real if particles aren't real

Serious question

>>8820699
Show me a set of -1 apples. Validity of mathematical concepts is not determined by whether they can be represented with apples

>>8820721
Funny, but actually the church was against the concept of infinity, because as they said, if we fill the universe with our infinity then there would be no space left for God.

Look up spinozas opinion on infinity,
He puts it nicely

>>8820803
Then there could be a set A such that A={1,A}, and that seems a bit counter-productive.
It doesn't make more sense then A={A}.

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>>8821912

>>8824323
If God was so great he could have created enough space in the universe for both.

>>8822528
>>8822546
Look up semiring theory. You just have a number for "everything beyond the arbitrary cutoff".

>>8820699
No there is a bigger problem with one set of infinities being bigger than another. It seems like it makes sense and doesn't at the same time.

>>8820685
OP has a point, there is no empirical example of anything infinite. I debated this online with someone ten years ago and they just kept begging the question over and over again like a little bitch.

>>8825890
how about the amount of stupidity of anons on this shit website?

>>8825890
Either the number of times you can add a natural number to 0 such that the result is still a natural number, or (if that's not empirical enough for you) the number of points between any two distinct objeccts.

If they object to the latter based on Planck length or quantum theory, then point out that the theory of quantum physics presupposes the natural numbers (you can't do quantum physics in ZFC-Axiom of Infinity) so if they believe in the validity of quantum physics then they must a priori believe in the existence of the natural numbers, so you can then take that as an example of infinity.

>>8826165
>Either the number of times you can add a natural number to 0 such that the result is still a natural number
This one is pretty silly, the number obviously doesn't exist. It can't be infinity, because by then you've reached infinity and infinity isn't a natural number thus contradicting the claim that it is still a natural number.
>the number of points between any two distinct objeccts.
And for this, the obvious response would be that mathematical points of no measure or size don't even begin to exist to begin with.

>>8826154
not an argument bitch.

>>8823720
>>8825316

AND THIS IS THE PROBLEM WITH MATHEMATICS, STUFF LIKE ``DIFFERENT SIZES OF INFINITY'' IS ACCEPTED WITHOUT QUESTION, EVEN THOUGH WHEN FIRST INTRODUCED TO MOST PEOPLE IT JUST MAKES 0 SENSE BECAUSE IT'S BULLSHIT

THE ONLY LEGIT ANSWERS DEFENDING THE AXIOM OF INFINITY IN THIS THREAD WERE

>BECAUSE IT PRODUCES SOME COOL RESULTS

OKAY NO ONE IS AGAINST THAT, BUT AN AXIOM SHOULD BE SOMETHING THAT'S CLEARLY TRUE AND THE AXIOM OF INFINITY ISN'T, MATHEMATICIANS USE IT WITHOUT AN ACTUAL REASON

>>8823764

OF COURSE YOU CAN'T PROVE SOMETHING THAT'S CRAP!

>>8820705
>IT STATES THAT THERE IS A SET WITH INFINITE ELEMENTS WHICH DOESN'T EXIST IN THE REAL WORLD
Spacetime is continuous and hence has infinite points.

>>8826744
NICE CIRCULAR REASONING

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>ctrl+f: 'real world'

>14 results

Holy... you really think it exists? I can't find it anywhere. All I find is more qualia, nothin' but qualia around. Oh, there's another one!

Infinity is proved you little bitch.

>he's this much of an engineering pleb that he thinks we need an empirical referent for the truth of mathematics
>dude what is set theory lmao

>>8826762
> axiom of infinity
> axiom
> proved
>>>/b/

>>8826165
I am requesting an empirical example of something infinite which means it needs to be physically observable or based on experience, I am not requesting a proof of infinity is based on theory or logic.

>>8826762

LOL

Why do people accept god?
It makes things easy just like infinity for mathematicians.

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>>8826767
>>8826777
>implying you can even think of a finite thing without involving an implicit infinity

Pic related is the only logic textbook you need.

>>8826782
So you're saying god is a mental construct?

>>8826762
> >he's this much of an engineering pleb that he thinks we need an empirical referent for the truth of mathematics
>dude what is set theory lmao

Nobody is saying you need infinity to be empirical for it to be used in mathematics retard, the whole point of this argument is to just point out that it is not empirically verifiable.

>>8826796
Where I come from 'empirical' denotes something bad, false, and unscientific.

>>8826794
Much like infinity a lot of people take god and use it for their own goals.
If a lot of people believe in something that doesn't make it real.

>>8826801
> Where I come from 'empirical' denotes something bad, false, and unscientific.

"Empiricism emphasizes evidence, especially as discovered in experiments. It is a fundamental part of the scientific method that all hypotheses and theories must be tested against OBSERVATIONS of the NATURAL WORLD."

>>8826801
Empiricism is the foundation of science you utter pleb

>>8824317
WOW YES EXCEPT THAT THERE IS NO AXIOM STATING THAT THERE EXISTS A SET WITH -1 ELEMENTS WHICH WOULD BE BULLSHIT, BUT THERE IS ONE STATING THAT THERE EXISTS A SET WITH INFINITE ELEMENTS WHICH IS BULLSHIT

>>8826857
Then it has some Weak foundations.

>>8826925
It's the only way for science. A priori knowledge is only possible in mathematics.

ALSO CANTOR KILLED MATHEMATICS TURNING THEM INTO A JOKE

I regret reading this thread, is your straight-jacket on CAPfag?

>>8827109
NICE ARGUMENTS BITCH

>>8822498
YOU REPRESENT REAL WORLD THINGS WITH ACTUAL NUMBERS, YOU CAN FIND ONE APPLE, TWO APPLES AND SO ON BUT NOT INFINITE APPLES

>>8826989
Mathematicians had been implicitly using infinity long before him. Cantor just tried to formalize the notion so we could do proper mathematics with it.

In this sense, Cantor is actually the unsung hero of finitism for giving us the tools to fully unveil the absurdities of the infinite.

>>8827174
YEA BUT CANTOR WAS THE FIRST ONE TO SERIOUSLY START SPERGING ABOUT INFINITE SETS, RESPECTABLE MATHEMATICIANS LIKE KRONECKER TRIED TO POINT OUT HIS NONSENSE. BUT TODAY THE SO CALLED MATHEMATICAL COMMUNITY ACCEPTS HIS CRAP AS A RELIGIOUS FACT. SAD!

>>8826958
And logic and philosophy

BUMP

COME AT ME INFINITISTS BOYS

>>8827452
As >>8822582 asked, do you accept the notion of pairs? Are there such things as pairs (4, 17) of natural numbers?

>>8827452
existence itself is the exploration of infinity
think about it for a minute, you'll realize its truth.

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>>8820702
What is entropy and heat death
checkmate

>>8827479
>>8822582
I BELIEVE IN THE EXISTENCE OF FINITE SETS, SO YES I BELIEVE IN A SET/PAIR OF TWO APPLES OR STONES

>>8827607
Cool. Yet you also seem to believe in a limited amount of natural numbers.

How many PAIRS of natural numbers are there? Is it larger than the amount of natural numbers? If so, is this a natural number itself, and how? If not, are there combinations of natural numbers that cannot form a pair?

>>8827620

>How many PAIRS of natural numbers are there?
UNDEFINED, INSTEAD OF TRYING TO SOLVE THIS KIND OF QUESTIONS USING INFINITY AND DIFFERENT SIZES OF INFINITY AND OTHER MEMES PEOPLE SHOULD CONSIDER TO JUST ACCEPT THATS UNDEFINED

AND THAT IS A PURELY THEORETICAL SITUATION NO REAL WORLD THING IS UNCOUNTABLE WITH A FINITE AMOUNT OF NATURAL NUMBERS

>Is it larger than the amount of natural numbers?
BOTH ARE UNDEFINED

>If so, is this a natural number itself, and how?
NO. IT'S UNDEFINED AGAIN

>are there combinations of natural numbers that cannot form a pair?
NO

>>8827639
but it isn't undefined, it's defined to be infinity :^)

>>8826701
Use of the term larger is a bit of memery. It's not literal.

>>8827868
"Meme" is such a magical word, it explains everything while letting you appear to have knowledge of what you're talking about!

>>8827889
It is tho.

>>8827889
Retard.

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>>8826857
>>Empiricism is the foundation of science you utter pleb
this is the faith of a 20 yo undergraduates in STEM

>>8820691
It's abstract which I don't think is wrong but have a hunch you might

>>8820685
In conclusion, op is a faggot

Doesn't this just count how many empty sets are in a set?

>>8828071
There is only one empty set as subset of another set (extensionality).

>>8828080
But looking at the formal def where
0={}
Then 1 = {0} = {{}}
Then 2 = {0,1} = {{}, {{}}}
It's kind of like counting the largest amount of layers deep you have to go into the set before you reach an empty set. I have some knowledge on formal logic and none really on set theory, but that's just a pattern i feel that occurs

>>8827915
?

>>8827209
>RELIGIOUS FACT
Hey man. You're welcome to your other opinions, but this is just bullshit. There's nothing religious about the axiom of infinity and modern set theory.

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>>8820685
Infinity is a trancendental concept that by design overreaches the Real. Noone is claiming that infinite sets exist in the real world, but they arrise as a natural trancentental extension of the finite sets.

This lies at the core of what mathematics is: we build transcendental models that encompass the Real, because many problems can be solved easier that way.

Take the natural numbers: in the Real we can work with sets with a number of elements and we can imagine that if we have a set with a number of elements that we can add another element; we get a hierarchy. Observe that there is no natural cut-off point for where the size of a set should end. (save for completely arbitrary limitations on how many elements you can write down) Given this insight into the nature of finite sets, we can make the trancendental step to the concept of the 'set of natural numbers' which contains this natural hierarchy of finite sets.

Although you may now think that this object is made up, note that it is not arbitrary, because it is precisely the object that captures the nature of what finite sets are and how they relate to one another: a hierarchy of objects without a natural cut-off point. The trancendental step just casts these features into a concrete object for us to study.

Read Kant nigger.

>>8828151
ur a cheeky kant m9

>>8820685
From your discussions, it seems that although you reject infinity, you do not accept a known, finite largest 'natural', either. Instead, you seem to claim that answer to questions about 'unbounded constructions' (x -> x+1, constructing pairs) have the answer 'undefined'. What then, is the practical difference between this 'undefined' and 'infinity'?

To me, it seems the difference is that 'undefined' means that somehow, at some point (unknown by definition?), the rules of extending constructings stop to apply and all the mathematics based om them stops.

This seems far more ridiculous than the notion of 'infinity', as you have made a lage amount of mathematics nondeterministic. Additionally, this behaviour has never been observed, so even empirically, this makes little sense.

Therefore, I believe that, given (potentially) 'unbounded constructions' (required to do interesting mathematics), we MUST be able to treat the number of objects we can construct as a proper mathematical entity. If this is not a natural number, then (excluding silly number hierarchies) we must consider the naturals unbounded, as otherwise even proofs involving only a finite amount of numbers first need to show the numbers aren't 'too large'. Currently, you have given no method for this and I doubt a sensible one can be created.

holy shit this is
>engineering
the thread

hurr durr this section of math doesn't directly related to real world its useless.
Stop it. You're worse than those kids in math class that always raised their hands and asked "When are we ever gonna use this teach/?? :^))"

>>8820699
Just because there aren't an infinite number of apples doesn't mean there aren't an infinite numbers of anything. What about you if you were counting with atoms? What about subatomic particles? What about dark matter particles?

It's absolutely possible that there are infinite numbers of yet undiscovered particles.

>>8827620
Capsboy BTFO. /thread

>>8827826
AND DATS THE PROBLEM WITH THE SO CALLED MATHEMATICIANS INSTEAD OF ASSUMING SOMETHING IS UNDEFINED THEY MAKE SHIT UP

>>8828145
IT IS LIKE A CULT, IF YOU TRY TO QUESTION THE AXIOM OF INFINITY YOU WILL BE ATTACKED AND SILENCED BY THE SO CALLED MATHEMATICAL COMMUNITY

>>8828161
I was going to write a reply complaining about the silliness of >>8827639, but your analysis seems to hit things on the nose much better.

Caps anon, please respond.

>>8828634
The whole point of something being undefined is that we are free to define it in a useful and mostly consistent way.

>one infinity can be bigger than other infinity
>set of infinity numbers

well what the fuck?
now sci will show that flies can generate 0.5kg downforce thrust

>>8828161
>From your discussions, it seems that although you reject infinity, you do not accept a known, finite largest 'natural', either.

SINCE I REJECT INFINITE SETS THE SET OF ACTUAL NATURAL NUMBERS MUST BE FINITE AND THEREFORE THERE IS A LARGEST ONE. AND ``NUMBERS'' GREATER THAN THAN ONE ARE MADE UP, THEY CAN BE STUDIED AND SHIT BUT THEY DON'T REPRESENT ANYTHING

AND THE LARGEST NATURAL, CALL IT FOR EXAMPLE IS A CONCRETE NUMBER BUT UNKNOWN/UNDEFINED

>What then, is the practical difference between this 'undefined' and 'infinity'?

THE DIFFERENCE IS THAT WITH UNDEFINED YOU DON'T ACCEPT THE NOTION OF INFINITE SETS SINCE THEY ARE AS IMAGINARY AS SETS WITH A NEGATIVE NUMBER OF ELEMENTS. INSTEAD OF MAKING UP A CONCEPT (INFINITY) JUST CONSIDER THAT IT IS UNDEFINED


>Additionally, this behaviour has never been observed, so even empirically, this makes little sense.

OF COURSE IT CAN'T BE OBSERVED, BECAUSE WITH A FINITE AMOUNT OF NATURALS EVERYTHING IN THE REAL WORLD IS REPRESENTED

>we must consider the naturals unbounded

TO CONSIDER THIS YOU NEED TO ACCEPT FIRST THE INFINITY AXIOM, IF YOU ACCEPT IT OF COURSE THE RESULTS WILL MAKE SENSE BUT THE AXIOM ITSELF DOESN'T

>>8828192

IT DOESN'T RELATE TO ANYTHING REAL, IT'S THE SAME AS STUDYING SETS WITH NEGATIVE NUMBER OF ELEMENTS. IF A MODERN DAY CANTOR DECIDED TO KILL MATHEMATICS EVEN MORE CLAIMING THAT THERE ARE SETS WITH A NEGATIVE NUMBER OF ELEMENTS, PEOPLE COULD DO NEW RESULTS CONSIDERING THAT AXIOM BUT EVERY SINGLE ONE OF THEM WOULD BE NONSENSE

SAME WITH THE AXIOM OF INFINITY

>>8828678
stop being retarded
you don't believe there's a largest natural number
you just believe the natural numbers don't form a set
say it like it is so you can act like the right kind of autist

>>8828400
>Just because there aren't an infinite number of apples doesn't mean there aren't an infinite numbers

TO CONSIDER AN INFINITE AMOUNT OF NUMBERS FIRST YOU MUST ACCEPT THE INFINITY AXIOM

>What about you if you were counting with atoms? What about subatomic particles? What about dark matter particles?

FROM WHAT WE KNOW THERE IS NO SINGLE EVIDENCE POINTING THAT THERE ARE INFINITE NUMBERS OF ANY OF THAT

>>8828695
NO NO NO NO

>>8828696
no matter if you accept or reject the axiom of infinity, there are still infinite objects that you have to deal with in math no matter what. you just claim that they aren't sets but they're still there

>>8828697
idiot. you don't know what you believe. you're just retarded. the axiom of infinity is equivalent to "natural numbers are a set"

>>8828696
>TO CONSIDER AN INFINITE AMOUNT OF NUMBERS FIRST YOU MUST ACCEPT THE INFINITY AXIOM
That is the thing, though. You don't.

The infinity axiom is one particular way to *codify* the *preexisting intuition* that there is an unbounded amount of numbers, not the other way around. It is not like mathematicians made up the axiom of infinity and thereby deduced that there are infinitely many natural numbers. Rather, there was the idea that the natural numbers are not bounded, and then the axiom of infinity was conceptualized as *one particular* way to represent this intuition in the language of set theory.

There being an unbounded amount of natural numbers was an accepted premise well before set theory was a thing at all.

>>8828678
>we must consider the naturals unbounded
>TO CONSIDER THIS YOU NEED TO ACCEPT FIRST THE INFINITY AXIOM
Not true
Unboundedness simply says that for every natural number you can find a natural number bigger than it.
You don't need to assume that the natural numbers form a set as all.

Consider the similar theorem in set theory that if x is a set then {x} is a set.
This means that sets are unbounded (more precisely, they are unbounded under quantification, see e.g. https://en.wikipedia.org/wiki/Bounded_quantification) but this does not imply the existence of the set of all sets, since that leads to Russell's well-known paradox.

>>8828701
LOL

CHECK YOUR FOUNDATIONS, THE AXIOM OF INFINITY COMES BEFORE THE CONSTRUCTION OF NATURAL NUMBERS.

NEXT

>>8828678
>AND THE LARGEST NATURAL, CALL IT FOR EXAMPLE IS A CONCRETE NUMBER BUT UNKNOWN/UNDEFINED
Ah, so the number of naturals is an entity that can be reasoned with in your framework, whatever you may call it.

This is a reasonable framework and very similar to those considered by 'ultrafinitists' (e.g. NJ Wildberger) It is most certainly 'non-standard' mathematics, but it is mathematics nevertheless.
It is less popular than assuming infinite cardinality of the naturals, mostly since the latter has let to more useful applications.

IMO, taking this alternative to the infinity axiom is a bit like using constructive logic instead of classical logic. You weaken your power to prove results, but get nicer proofs or axioms. (Note that I personally prefer the axiom of infinity, simply for the conceptual abstraction it gives, but I can understand those that feel that mathematics must be shackled to 'the real world' somehow, as it is produced by fallible, finite, humans)

>THE DIFFERENCE IS THAT WITH UNDEFINED YOU DON'T ACCEPT THE NOTION OF INFINITE SETS SINCE THEY ARE AS IMAGINARY AS SETS WITH A NEGATIVE NUMBER OF ELEMENTS. INSTEAD OF MAKING UP A CONCEPT (INFINITY) JUST CONSIDER THAT IT IS UNDEFINED

The two concepts still seem isomorphic to me. It still seems like using a different word for the same concept.

>TO CONSIDER THIS YOU NEED TO ACCEPT FIRST THE INFINITY AXIOM, IF YOU ACCEPT IT OF COURSE THE RESULTS WILL MAKE SENSE BUT THE AXIOM ITSELF DOESN'T
I meant say that unless we explicitely bound all recursively defined 'expanding' constructions (such as constructing x+1 from x), rejecting infinity is inconsistent. But if we add an explicit bound as an axiom, such recursions without a 'last case' bounding them would not be well defined, in which case there is not problem.

>>8828702
>It is not like mathematicians made up the axiom of infinity and thereby deduced that there are infinitely many natural numbers

BUT THE CONSTRUCTION OF THE SET OF NATURAL NUMBERS (WHEN CONSIDERED INFINITE) NEEDS THE AXIOM OF INFINITY TO WORK. THE AXIOM OF INFINITY COMES BEFORE ANY MENTION OF NATURAL NUMBERS AND CLAIMS THAT THERE ARE INFINITE SETS

>>8828704
OKAY I READ WRONG THERE

>>8820685

94% of mathematics is a simple substitution operation. the subject substituted is a various interdependent "progression" which could be described as lengthy episodes of sub-actions.
Because it's impractical to always enumerate all those sub-actions those are substituted for symbolic references.
Normie brain does this 101% of time. They have turboboosted semantic memory and lack havily the sticky focus to iterate the episodic memory. And us math aspies just can't stand the working with all the sub-actions some other aspie enumerated on paper due to occasional attention deficits - the impatient impulsive ones when the focus shifts and sticks to the steps past the result prior verification. So we complain but in the end accept it.

The remaining 6% of math toolkit is the zillion other methods used only occasionally. Too long to list.

>>8828719
>BUT THE CONSTRUCTION OF THE SET OF NATURAL NUMBERS (WHEN CONSIDERED INFINITE) NEEDS THE AXIOM OF INFINITY TO WORK.
No, it doesn't. That's one particular way to construct them out of many. There are many formalizations of the natural numbers that do not mention infinity at all.

Yes, SET THEORY starts with infinite sets and then defines natural numbers relative to those. But that is because set theory is about sets first, and numbers second. Theories that focus on numbers don't need to bother with infinities at all.

>THE AXIOM OF INFINITY COMES BEFORE ANY MENTION OF NATURAL NUMBERS AND CLAIMS THAT THERE ARE INFINITE SETS
Not in Peano arithmetic, for example. Peano arithmetic defines unboundedly many natural numbers, without talking about infinite sets at all.

I am not sure whether you realize this, but it is not like set theory is the foundation of mathematics. It is AN attempt at a foundation of mathematics, not THE foundation; there are axiomatic systems that talk about infinitely many things without ever mentioning infinity.

>>8828705
>I am stupid
the natural numbers are the equivalence classes of the equivalence relation of the existence of a bijection for the classes of finite sets. a finite set is one where there is no injection from it into itself. you don't need infinity for any of these, you're just retarded.

>>8828727
What are you even trying to say? That induction is pointless?

>>8828719
the axiom says the infinite class of finite ordinals is a set
you don't need an axiom to get an infinite class, as soon as you can generate arbitrarily large finite sets

What you are proposing is called finitism and it's respected as an opinion in the philosophy of mathematics.
But actually it's not very feasible to leave it away, because
- calculus won't work anymore (no infinite series means no limits)
- physical models won't work anymore, because they are based on calculus usually
- measure theory won't work anymore, because you can't have any σ-algebras with infinite sets anymore
- no probability theory because it's based on measure theory
- and so on

I don't see why we should restrict ourselves like that only because there's no evidence that infinity exists in the real world. It's handy, so, yeah...

And actually there are a LOT more fundamental problems with maths than this one axiom. For example the fact that all axiomatic systems are either incomplete or contain contradictions (aka maths is fundamentally flawed), so there's no use in taking it seriously anyways

>>8828793
but you wont need any of that anymore because all of that can be approximated :^)))))

>op can't into abstraction
Who do we blame for this?

>>8828733

>IMO, taking this alternative to the infinity axiom is a bit like using constructive logic instead of classical logic. You weaken your power to prove results, but get nicer proofs or axioms. (Note that I personally prefer the axiom of infinity, simply for the conceptual abstraction it gives, but I can understand those that feel that mathematics must be shackled to 'the real world' somehow, as it is produced by fallible, finite, humans)

YES DATS IT, FINALLY SOMEONE THAT GETS IT

>The two concepts still seem isomorphic to me. It still seems like using a different word for the same concept.

NO. THERE IS A DIFFERENCE BETWEEN STATING THAT SOME SET HAS INFINITE ELEMENTS, AND STATING THAT THE NUMBER OF ELEMENTS IS NOT INFINITE BUT UNKNOWN/UNDEFINED

>>8828729
OF COURSE, BUT I'M TALKING ABOUT ZFC CONCRETELY. OF COURSE THERE ARE OTHER FOUNDATIONS OF MATHEMATICS. BUT ZFC IS THE STANDARD ONE IN THIS DAY AND IN THIS THREAD I POINT OUT ONE OF ITS AXIOMS

>>8830016
MEANT TO QUOTE >>8828708 INSTEAD OF >>8828733, BUT NOW ITS UR TURN

>>8828733
>I DONT UNDERSTAND THIS TOPIC

COOL BUZZWORDS BUT THIS IS ABOUT ZFC

I SAID NEXT

>>8830016
>OF COURSE THERE ARE OTHER FOUNDATIONS OF MATHEMATICS. BUT ZFC IS THE STANDARD ONE IN THIS DAY
Really? In practice mathematicians seem to use some kind of dynamic type theory, with ZFC being only a convenient alibi for the rigorousness of their work. If ZFC was really the foundation of math you wouldn't see refutable statements like [math]\mathbb{N} \subseteq \mathbb{Z} \subseteq \mathbb{Q} \subseteq \mathbb{R} \subseteq \mathbb{C}[/math].

I personally subscribe to the theory of meme arrows, since it allows you to work with the natural numbers as a finite object without worrying about size issues (see e.g., https://en.wikipedia.org/wiki/Natural_number_object).

>>8828708
Could I ask you a question. I want to ask Wildberger or someone, but I'm too lazy to go there personally.

In normal mathematics, I can apply the existential and universal quantifiers over infinite sets. Ex:
(∀n∈ℕ)(n + 2 = 2 + n)

I like this nice, simple, and /symbolically formal/, way of expressing a statement. How do these ultrafinitists respond? How can I formally express such ideas without quantifying over infinite sets? Is there an alternative? What alternative do they have? I suppose they might have one, but I don't know.

>>8830331
>refutable statements like N⊆Z⊆Q⊆R⊆C
What?

>>8830598
In ZFC (this is important):
- N is defined as 0 = {}, n+1 = n union {n}.
- Z is defined as a set of ordered pairs of natural numbers (n,m) satisfying certain conditions
- An ordered pair (n,m) is defined as { {n}, {n,m} }.

Lemma: Every ordered pair has cardinality 2.
Lemma: Every natural number n has cardinality n.
Therefore any natural number that has cardinality other than 2 cannot be an ordered pair, and in particular, it cannot be a member of Z.
Therefore [math]\mathbb{N}[/math] is not a subset of [math]\mathbb{Z}[/math] in ZFC.

Similar arguments can be made for the rest.

>>8830609
>Lemma: Every ordered pair has cardinality 2.
...and I forgot (again) that (n,n) has cardinality 1, so the technically correct result should be that no natural number greater than 2 can be an element of [math]\mathbb{Z}[/math].

See this is why I prefer category theory to set theory (even if it's a bit of a meme); you're far less likely to make type errors like these.

>>8830614
IIRC, technically games have to be played in pure set theory in order to construct pairs, because pairs are not basic things in ZFC.

Thanks for the clear answer. I am, of course, unimpressed.

ya'll niggas need potential infinity

>>8830618

For a moment I thought you were referring to combinatorial games in the sense of Conway, which are indeed defined as pairs in his seminal work "On Numbers and Games". Funnily enough, near the end of the book he rails against set theory for the same reason that I do.

I don't have anything against set theory though, it's just not my first choice of theory for axiomatizing mathematics. After all, problems like [math]\mathbb{N} \subseteq \mathbb{Z}[/math] are correctible in set theory by replacing [math]\subseteq[/math] with an appropriate injection satisfying some characteristic property instead, but why build with shaky foundations and spend time propping them up ad-hoc, when you can just make sure your foundations are solid from the beginning instead?

Also, though I'm not an ultrafinist, I'll try and give my response to your question >>8830590
In category and type theory, mathematical statements are not endowed with any semantic interpretation but are treated purely syntactially as formal strings of symbols, and mathematical proofs are treated as meaningless games of symbol manipulation. Your example proposition
(∀n∈ℕ)(n + 2 = 2 + n)
would translate into a theorem asserting the associativity of arrow composition, and it would be verified by substituting n,2 with their appropriate definitions and appealing to the universal property of N (as alluded to in >>8830331) to justify the equality.

So I'd say (and again, I'm not an ultrafinist so others may have different opinions) you can keep your symbols (in fact you can keep whatever notation you're the most comfortable with), just avoid wading into questions over "is N infinite" or "how many elements are in N" since that's a model-dependent question with no correct answer that can only lead to pointless non-refutable shitflinging.

>>8830609
>>8830614
how silly
anyone taking a first course in set theory will notice that after you build Z, you identify N with the corresponding subset of Z and so on, you finally identify them all as subsets of C

this is done without mention in most constructions: you don't care about objects as sets, but as isomorphism classes

>>8830609
Or
In ZFC
- N^* is defined as 0 = {}, n+1 = n union {n}.
- An ordered pair (n,m) is defined as { {n}, {n,m} }.
- Z^* is defined as a set of ordered pairs of N^* (n,m) with an equality relation (a,b)~(c,d) <=> a+d = b+c
- Z is the equivalence class N x N / ~
- N = {x in Z : exists (a,b) \in x such that b = 0 }
might have fucked up that last pert. too much tequilla.

You can show there's a bijection between N^* and N, and so they have same cardinality
And directly from the construction of N shows that it is a subset of Z

similar arguments can be made for the other sets so you end up making statements like N⊆Z⊆Q⊆R⊆C make sense.

>>8830643
>finally
What if you want to work with the quarternions? What if you want to apply the Cayley-Dickson construction an unbounded number of times?
Will you ever be able to define the natural numbers in that case?

> you don't care about objects as sets, but as isomorphism classes
Exactly. So why not axiomatize algebraic structures directly?

>>8830642
>mathematical proofs are treated as meaningless games of symbol manipulation
Note that this is the general scheme that I subscribe to.

>>8830650
> So why not axiomatize algebraic structures directly?
the way i see it (not a professional mathemagician of any sort), you have to show that those isomorphism classes are non-empty before manipulating them.

That being said, I don't get the first order logic/ZFC stuff. Seems a bit too hand wavey on why things are what they are when you get that low, but wikipedia might not be the right place to learn these things.

>>8830650
>Exactly. So why not axiomatize algebraic structures directly?
I don't have enough exposure to such schemes to comment beyond wild guesses. My wild guesses is that using ZFC allows keeping fundamental principles to a minimum, and providing a single basis for interpretation of second-order logic statements like:
(∀n∈ℕ)(n + 2 = 2 + n)

I just find it intuitively hard to express that without referring to sets, e.g. understanding that set of symbols as meaning "for all elements of this set, property P holds", e.g.
(∀x∈S)P(x)
and I find it intuitively hard to express this without recognizing that ℕ is not finite, e.g. infinite.

>>8830660
Correction: This is just a first-order logic statement, right? Apologies. I don't claim to be an expert here.

>>8830660
Disclaimer: I'm not a professional mathematician either, everything I know about foundational mathematics is entirely self-taught.

>you have to show that those isomorphism classes are non-empty before manipulating them.
Not sure what you mean by this. Surely any meaningful isomorphism class that you construct is going to be non-empty because you know what goes into it?

>>8830660
And I'd say that interpreting meaningless strings of symbols is a fruitless endeavour: [math]\mathbb{N}[/math] is not a set in the informal English sense, but just a mark on a piece of paper. It's definitely not a position that I expect people to be comfortable with.
(And yes, that is a first-order statement.)

>allcaps
saged and hidden

>>8830670
Specifically, I regularly do things like:

Let X={11, 12}
Let P:X→boolean, P(x) = (x+2=2+x)

11+2 = 13 = 2+11
⇒ 11+2 = 2+11
⇒ P(11)

12+2 = 14 = 2+12
⇒ 12+2 = 2+12
⇒ P(12)

⇒ P(11) ∧ P(12)
⇒ (∀x∈X)(P(x))
⇒ (∀x∈X)(x+2=2+x)

The rules for the symbol manipulation of the "∀" symbol is defined for finite sets in this fashion. I'm not sure how else you might define the rules for symbol manipulation without implicitly referring to sets.

To put this another way, consider the following two claims:

(∀x∈X)P(x)
(∀n∈ℕ)P(n)

The symbol "∀" has the same "meaning" in both claims. In particular, one can apply the rule of "universal instantiation" for both claims. The rule of "universal instantiation" loosely is given a true expression of the form "(∀x∈X)P(x)", one may logically derive the expression "P(x)" for any element of X at the writer's discretion".

In other words, I don't know how you can do the same sort of math and logic that I'm familiar with, without the rule of universal instantiation, and I don't know how one can talk about the rule of universal instantiation without referring to sets, and most importantly I don't know how one can invoke the rule of universal instantiation when quantifying over naturals without having an explicit symbol "ℕ" that refers to the set of the naturals.

I suppose that you don't need to answer the question "does this symbol ℕ refer to an infinite set?", but that seems like an immediate and obvious question to ask. I can ask "What is the size of the set X = {11, 12}?" and the answer is "2". Similarly, I should be allowed to ask "What is the size of the set referred to by the symbol ℕ?".

>>8830590
>(You)
Although >>8830642 has some good points, II'd like you to note that there does exist a simple well-defined and provable analogue of the statement
(∀n∈ℕ)(n + 2 = 2 + n) in Wildberger's framework.
Wildberger is a bit different from most ultrafinitists in the sense that he explicitely defines a largest number and matches it to a physical constant, while others make an effort to avoid this situation.

We can interpret your statement as over the _finite_ naturals as defined in Wildberger's framework. In that case, there are no infinite sets in your statement, although we might wish to use different symbols to avoid confusion.

There is new problem, however. The expression n + 2 is not well defined, as if we take one of the largest 2 numbers for n, adding 2 no longer results in a number. So, we have to restrict the statment a bit. Define ℕ[-x] for all x∈ℕ as the subset of ℕ minus the largest x numbers from ℕ. The statement (∀n∈ℕ[-2])(n + 2 = 2 + n) is now well defined and is provable.

If we wish to do something more interesting and define associativity of +, we must first define all pairs a,b∈ℕ for which a+b is well defined (i.e. the result is not too large)
and then show that a+b = b+a for all pairs in that set.
Although this set has finitely many elements, it is larger than that of ℕ, so we cannot count its elements. It seems that rejecting infinity bites you in the back eventually...

Note that this is only my personal interpretation of Wildberger's framework, so he might have a more satisfying resolution to practical proving within his framework than I do.

>>8830983
>Wildberger is a bit different from most ultrafinitists in the sense that he explicitely defines a largest number and matches it to a physical constant, while others make an effort to avoid this situation.

Thanks.

W.r.t. Wildberger, I'm just shaking my head. It's so silly.

>>8828696
The universe is expanding faster than the speed of light, moron. That means if you travel from point A to the edge of the universe at the fastest possible speed, it will take infinite time to reach there.

How is this such a difficult concept for you to understand?

>>8830590
>(∀n∈ℕ)(n + 2 = 2 + n)

CONSIDERING A LARGEST NUMBER IN THE SET OF NATURAL NUMBERS AS I SAID BEFORE AT SOME POINT THE ``NUMBER'' n + 1 BECOMES NONSENSE LET ALONE n + 2

AND BECAUSE IT ISN'T AN ACTUAL NUMBER (UNDEFINED) YOU CAN'T USE THE TRADITIONAL OPERATIONS OR EQUALITIES OR WHATEVER

SO THE STATEMENT

(∀n∈ℕ)(n + 2 = 2 + n)

IS FALSE ASSUMING A LARGEST NATURAL NUMBER. AS OTHER ANON SAID TO MAKE IT TRUE YOU NEED TO EXCLUDE THE TWO BIGGEST NATURALS FROM ℕ

>>8820778
Fuck you

>>8831016
>it will take infinite time to reach there.

IF YOU CAN'T GET THERE THAT DOESN'T MEAN IT TAKES AN INFINITE AMOUNT OF TIME LOL. IT JUST MEANS U CAN'T

>>8831865
>there is a biggest natural number
Lol.

>>8831865
Also, dude, you cannot even do simple things like prove the fundamental theory of algebra.
https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra

Similarly based on the intermediate value theorem:
https://en.wikipedia.org/wiki/Intermediate_value_theorem
I know that if some continuous function has a value P above the line x = 0, and elsewhere has a value Q below the line x = 0, then there is a point where the function crosses the line x = 0, e.g. there exists a value x so that f(x) = 0 and P < x < Q. This is another thing that you cannot prove.

How am I supposed to take you seriously when you cannot prove things like this that are common in high schools everywhere?

>>8832077
>How am I supposed to take you seriously when you cannot prove things like this that are common in high schools everywhere?
>implying anyone gives a shit if you take them seriously.

You're a retarded joke guy.

>>8820699
count for the rest of your life, you still won't be done

>>8832077
>How am I supposed to take you seriously when you cannot prove things like this that are common in high schools everywhere?

This is an unfair requirement for alternative proof or axiom systems, as they intentionally limit their power to proof results to get different properties on their proofs.

The simple statement ~(~A /\ ~B) -> (A \/ B) cannot be proven in intuitive logic, it requires the law of excluded middle ( A \/ ~A ).
Does this mean we shouldn't take constructive mathematics seriously? I think not, as proving everything the 'usual' mathematics can is not the goal here.

File: 1474428802006.png (134KB, 334x393px) Image search: [Google]

134KB, 334x393px

>>8820699
>it's another "brainlet assumes time is finite" episode