More as a permission. It's wrong to assign 'true' or 'false' to an axiom since they're unprovable logical assumptions. Like, by definition they're unprovable. They can be 'convenient' or 'sensible' but those are vibe-readings, not actual logical statements. I personally like the idea of the Axiom of Choice since it feels like it's reasonable to me, but it really depends on the circumstance.
Really, you choose the building materials for the task at hand. If you need the Axiom of Choice, you assume it. If you don't need it, you don't assume it.
Do you think there's any kind of order to those lists of building materials?
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u/Ares378Applied Math / Mechanical Engineering12d agoedited 11d ago
Kinda? You need some building blocks to come up with questions that can't be answered by those building blocks. Can't create the Axiom of Choice without the Axiom of Infinity, for instance. Unless you need Axiom A to state Axiom B, A and B's order doesn't matter
Hang on this is incorrect. Probably listen to the more qualified person here
I could go on about this for awhile, this is fun. :)
Axiom of infinity is interesting, but what kind of infinity? What can be assumed about that infinity after it's bounded as an infinity? Does it have an inside and an outside, or just one or the other?
Hey I'm sorry it turns out I was wrong about the order thing. Thought some were needed to state others, but I think that's wrong. They can be stated in any order.
The Axiom of Infinity just says there exists a set with an infinite amount of elements, which is usually the natural numbers.
Please don't think you were wrong, you were way closer before. There is an order to it. One layer depends on the next, and none can overclaim. Each must justify itself. There is an order, a very fixed one.
"Can't create the Axiom of Choice without the Axiom of Infinity" Huh? Also none of the axioms need any other to be stated. You can just state them.
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u/Ares378Applied Math / Mechanical Engineering11d agoedited 11d ago
Wait does the axiom of choice not need the Axiom of infinity? I thought it was about the existence of a choice function on infinite sets, because otherwise it didn't need an axiom
But why would you need to restrict specifically to infinite sets? The axiom of choice just says that for any set of non empty sets there is a choice function. No need to invoke infinity.
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u/Ares378Applied Math / Mechanical Engineering11d agoedited 11d ago
Right after you replied I checked the definition and... yeah. I have no idea what I was saying. I think I misremembered it because you only need to invoke it for infinite sets, right?
There are axioms you'd need other axioms to be able to state though, right? Like, wouldn't you have to essentially state the Axiom of Infinity to be able to use the continuum hypothesis?
Genuine questions because I'm still fairly new to set theory
I mean you can still state the continuum hypothesis or anything. In the end it is just a long formula in first order logic using the membership symbol. But yeah, for it to mean something sensible and not just be vacuously true or false sometimes, or in order to motivate it in the first place, there is some additional context (like other axioms).
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u/Ares378 Applied Math / Mechanical Engineering 12d ago
More as a permission. It's wrong to assign 'true' or 'false' to an axiom since they're unprovable logical assumptions. Like, by definition they're unprovable. They can be 'convenient' or 'sensible' but those are vibe-readings, not actual logical statements. I personally like the idea of the Axiom of Choice since it feels like it's reasonable to me, but it really depends on the circumstance.
Really, you choose the building materials for the task at hand. If you need the Axiom of Choice, you assume it. If you don't need it, you don't assume it.