>open up a book on mathematical foundations, eager to learn

>>8646911
>foundations on mathematics

Which foundations? If sets are good enough then Set Theory by Jech will probably suffice. Other than that I've a few people rave about Homotopy Type Theory (which is available online), never read it myself so can't really comment on it.

File: ha.jpg (15KB, 500x375px) Image search: [Google]

15KB, 500x375px

>>8646911
>most rigorous
>unambiguous
>exact explanation

You've already failed at anything /sci/ if those's aren't either givens, implied, or the antonyms aren't applied.

Be a wolf not a sheep.

>>8647008
Thank you for your 'feeling'/'intuitionistic approach'

But that will definitely NOT help answering actual questions like machine learning and AI

I'm sure your approach is excellent in fields like economy and marketing though, but not mathematics and programming

>>8647084
Set is a primitive. You can't define it rigorously. This doesn't matter for computation since computers don't have to understand what the primitive is to work with it.

>>8647131
When baby is born, does it know what a set is?
How about bacteria?
How about molecule?

What information you need to give a molecule to make it understand the "primitive" concept of set? I believe molecules don't understand the concept that 'some things belong together in some kind of group'.

I'm not sure if baby understands the concept of set, because DNA could have 'instructions' for it. But what I do know is that he will eventually 'learn' the concept.

Same with me. I remember when I didn't understand the mathematical concept of set. I just received years of ambiguous information and eventually got the hang of it. Now from that 'ambiguous' information, remove all excess and what is left?

http://www.math.toronto.edu/weiss/set_theory.pdf

File: Asian_Bus_Girl.jpg (118KB, 1280x960px) Image search: [Google]

118KB, 1280x960px

What's you background, OP?
Do you know predicate logic already?

Sets are defined in set theory, which is a theory written down in a predicate logic (a formal language), and the definition is an implicit one.

Consider the explicit definition
[math] x_f = 3+2 [/math]
There are rules to reduce that right hand side expression, and you'd end up with 5.
On the other hand, an implicit definition is something like

[math] x_o = \sum_{k=1}^\infty \dfrac{1}{2^k} [/math]

or

[math] x_r = \sum_{k=1}^\infty \dfrac{1}{k^2} [/math]

Because given a sequence [math] (a_k)_k [/math] like those implied in the definitions above, the expression "[math] \sum_{k=1}^\infty a_k [/math]" has no direct terminating computational content, like +.
Because, in reality, the formula
[math] \sum_{k=1}^\infty a_k = y [/math]
is a creature of analysis and is short for
[math] \forall (\varepsilon\in{\mathbb R}_{>0}).\,\exists (m\in{\mathbb N}).\,\forall (n\ge_{\mathbb N} m).\,| \sum_{k=1}^n a_k - y \, |<\varepsilon [/math]
and nothing else.
The above is a predicate in y, you may denote it by P(y), and so
[math] x_o = \sum_{k=1}^\infty \dfrac{1}{2^k} [/math]
really reads Q(y) for some predicate Q, and now there is the task to question what number - if any - fulfills this property. In that case, 1/2+1/4+1/8, the number y=1 fulfills the predicate
[math] \forall (\varepsilon\in{\mathbb R}_{>0}).\,\exists (m\in{\mathbb N}).\,\forall (n\ge_{\mathbb N} m).\,| \sum_{k=1}^n \frac{1}{2^k} - y \, |<\varepsilon [/math]
and it's the only number which does that. Thus this Q gives an implicit deifnition of the infinite sum being 1.
And for the other sequence with 1/k^2 there is no natural number y which fulfils the predicate. That infinite sum is actually "pi^2 / 6", although the very definition of the reals is also not constructive itself.

A set is implicitly defined by the axioms of set theory. They are like a bunch of formulas with y's, and then you can work with the formulas alone.

>>8647084
>>8647144
You're retarded. ZF(C) is axiomatic

File: images.png (4KB, 281x179px) Image search: [Google]

4KB, 281x179px

>>8647189
Let me give you another example, one which smells like algebraic geometry.

Consdier the expression
[math] s(x,y) := x^2+y^2-2^2 [/math]

Now the set
[math] S = \{ (x,y) \ | \ s(x,y)=0 \} [/math]
are the points in R^2 which are of Euclidean distance 2 away from the origin. If you draw that, you'll see a circle in front of you.

Similarly, the expression
[math] t(x,y) := x^2-2 x+y^2-2 y-47 [/math]
(which you may write also as [math] (x-1)^2+(y-1)^2-(8-1)^2 [/math])
becomes a circle of radius 8 located around the center point (1,1) on your piece of paper.

Now consider the set
[math] T = \{ (x,y) \ | \ x^4-2 x^3+2 x^2 y^2-2 x^2 y-51 x^2-2 x y^2+8 x+y^4-2 y^3-51 y^2+8 y+188=0 \} [/math]
I generated that expression by multiplying s(x,y)·t(x,y). Since s(x,y)·t(x,y) is 0 whenever either of those two factors is zero,
you can see that the set T is exactly the points on the figure with two circles. And in a similar way, you could get the set in pic related. It's pretty easy now to capture pictures of any number of circles with equations.

Algebraic geometry works with polynomials and "speaks of circles", but in fact to do the math of finding zeros of polynomials, you don't need "circles" as you know from real life - while still being able to answer questions about them, after you translated them.
You can be happy or not about a definition like "a circle 'is' a set of points..." but it doesn't matter. For algebraic geometry, that's it. For synthetic geometry, that's Decartesian plebbery. A circle isn't restricted to the idea of one or two mathematicians.
You can add circle pictures now by multiplying polynomials, very much like set theoreticians specify set unions and intersections by writing down conjunctions and disjunctions of fomulas.
And set theoreticans work with their formulas (talking about "sets") just like algebraic geometers work with their polymonials (talking about "surfaces")

>>8646911
Hi, instead of learning math I just read 4chan and wikipedia.
>logic master after 1 year

Played 1 day with wolfram

> wolfram can't comprehend logic and only gives a positive as an aproximation. And a negative as a strange fraction of 1.

What now?

>>8647189
>>8647238
This was an interesting read, thanks!

I think the
>a set is implicitly defined by the axioms of set theory. They are like a bunch of formulas with y's
rings a bell for me.

I'm studying physics and recently been studying number/set theories and some predicate logic but most texts I read I just disagree them too hard.

>>8646911
Rigorously defining what a 'set' is inevitably leads to unsatisfactory answers. You have to resort to some metalanguage or use a deflationary definition of sets as atomic objects.

To unambiguously define a set in such a way that a hypothetical alien would understand is to circumvent the fact that human knowledge is ultimately subjective (the product of information filtered through a specifically human mind). The definitions in a dictionary are ultimately circular (how do you define words without words?), and yet we all know what they mean. The problem ultimately boils down to the problem of language and how it is acquired.

Well in ZFC for example, what we call "set" are not really "set" they are just object of the theory and we define operations like "in" but it is not really a "in" operations, but it is defined so it match with our intuition

>>8647084
>machine learning
>AI
That's so far from set theory it's not even funny. Though you can argue that a big part of ML is statistics which is basically applied measure theory it's beside the point. Read Mendelson's book on logic of you want foundations, though. It's good.

>>8647084
>>8647434
P.S. Don't use the word "intuitionistic" in that wat. It means a very different thing in math.

>>8646911
How many fongurs does the alein have because that's how many base numbers he should use like if he only has 1 fongur he should learn math in binary.