Legit problem with the axiom of infinity

>>9102538
If you think it can be proven from the other axioms, then show us the proof.

>>9102538
People have considered this sort of scheme, I think you'll find this article very interesting
https://www.quantamagazine.org/mathematicians-bridge-finite-infinite-divide-20160524/

>>9102538
Because the existence of infinite sets can't be proven with the other ZF axioms. Think for two fucking seconds before asking a stupid question, and post it in the stupid question thread.

>>9102674
That's not what he's asking, retard. Read the post before replying.

>>9102679
Yes that's what this retard is asking. "Is there another simpler axiom that don't say 'an infinite set exists' but implies it ?". The answer is fuck no, if there was it would either be implied by the other axioms of ZF, impossible since what I said in the other post is true, or be less intuitive in which case his question is still dumbfucking stupid.

>>9102682
>be less intuitive
citation needed

>>9102682
>>9102674
You are legit retarded because I am not asking to prove infinity with the other ZF axioms, I am talking about replacing the axiom of infinity with a new axiom that is equivalent to the axiom of infinity but does not directly mention infinity itself.

It is trivial to find a statement that is equivalent to the axiom of infinity, but then the goal would be to find the most elementary or the most intuitive one.

what do you propose?

i don't think it's possible to prove the existence of infinite sets without an axiom to guarantee they exist in ZF at least

>>9103106
>i don't think it's possible to prove the existence of infinite sets without an axiom to guarantee they exist in ZF at least

Yeah. I simply say that we should find an axiom equivalent to the axiom of infinity, but that does not directly construct an infinite set like the axiom of infinity does.

>>9102543
He's not saying that it can be proven from the other axioms. He's saying that as an axiom, it's too "big" (for lack of a better term) compared to the others. And the reason for this "bigness" is that infinity is a concept constructed from "smaller" entities like "integers", "successors", etc. The idea of infinity didn't just come out of the sky, it has to reference other mathematical concepts to be explained and it's thereby weird to be put as an axiom alongside the rest of ZF which are self-explanatory.

>>9102674
Sure, you must posit an infinite set apart from other ZF axioms...but the question then becomes why?

Mathematical axioms are supposed to be the very starting ground of the whole system, right? They're the entire foundation for what comes afterwards. So imagine that you had just declared all the other ZF axioms except for infinity. What would make you want to declare that "there is an infinite set"? Where would such an idea come from? It's not like there's some obvious gaping whole in ZF that's going to prevent practical math from being done. Declaring the axiom of infinity is pretty arbitrary and could have weird repercussions down the road.

>>9103126
>I simply say that we should find an axiom equivalent to the axiom of infinity
WHY, though, should we do this? Really think to yourself: "Why do I care whether the axiom of infinity is declared or not?"

The answer is likely to be "My teachers taught me that infinity exists and it would be weird if that concept didn't make it into math." If that's the case you're just perpetuating an existing idea without really understanding what it means or the ramifications it could have. You really need to step back and think why it's so essential to have such an idea declared as an unassailable, foundational idea in math.

It's strange to my mind to think that there are people out there who want to declare the existence of an object which they don't understand and have never experienced. Consider the following: it's impossible to construct an infinite set from the other axioms of ZF. Have you considered that this could be, in fact, an indicator that we shouldn't be messing around with infinities, rather than declaring more axioms to shoehorn them in?

>>9102538
>Why hasn't this been done before?
Because there's no way to get to infinity dealing only with finite operations on finite sets.
Any axiom you introduce that is sufficient to prove N exists will by necessity have some sort of statement about infinity embedded in it already, and at that point it's more complicated than just saying "there exists an infinite set".

>>9103416
I think you are just lazy. I recommend you also read the article posted above. It is about the recent research about how to prove infinite statements with only finite ones.

It seems like it can be done, and if it can't then at least we should find a good answer for why that is. Just ignoring it is pretty lazy and against the spirit of math.

>>9103423
I actually agree with you more than you might think.

If it's possible to prove infinite statements with only finite ones, I won't have as much of a problem with those infinite statements. My beef is with declaring infinity as an axiom. That's not ignoring infinity. If anything, declaring the axiom of infinity is "being lazy" more because it's explicitly assuming that there are no problems with infinity and that it's understood enough to have utility. That's more arrogant to me than stepping back and saying "hey, before we set up the pillars of this field let's make sure we really know what we're declaring."

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613KB, 498x498px

>>9103418
>there's no way to get to infinity dealing only with finite operations on finite sets.

>>9103423
>recent research about how to prove infinite statements with only finite ones.

>>9102538
I worked with set theory for a while so I will try to explain it in simple way.

Maybe you should try and focus your attention on what is it that other axioms of ZFC are saying. They are designed so that the set theory can show us what kind of collection do we define to be sets. They don't give us precise definition of set but are allowing us to say "Hey, this IS a set". So if we are given a set or two, we can make set that consists of union of those sets, pairs of sets, power set, even subset. They even allow us (by axiom of union) to consider elements of elements into new set.

Problem with this (as I am sure you understand) is that they cannot produce set that has "more complicated structure" (even axiom of power set is not that strong in case of finite sets). We can make sets with more elements but none of them can be much larger than original sets. In this approach we cannot even consider sets of all natural numbers even though we can create each of those numbers. Existence of infinity set is given by beautiful axiom of infinity that says that there is a set which contains empty set as elements, and for every set x belonging to it, it takes {x} to be its members.

If you followed correctly definition of natural numbers, then your intuition should be more than enough to realize that this axiom gives us set of all natural numbers. Existence of this set cannot come from other axioms.

On the other hand, you may ask: What about the empty set? How can we say that it is a set containing no elements? When we write it in first order logic, it just says that for any set X, set X doesn't belong to empty set. Where do these sets "X" come from, and shouldn't there already be infinite amount of them lurking in some universe V since we can examine each of them?

>>9103126

It would be wrong to find axiom that is equivalent to it because then it doesn't make sense not to take this pretty intuitive approach (via axiom of infinity). On the other hand, if we take axiom which implies axiom of infinity, we would have stronger axiom which is, again, not the point, as you said in OP.

>>9103502
...I forgot to add, If you want equivalent definitions of this axiom, or stronger ones that can replace it and who sound more interesting to you, check Fraenkel - Foundations of Set theory, p 44. There are like three types of this axiom. The one accepted is the most natural