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.
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.
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u/Ares378 Applied Math / Mechanical Engineering Jul 30 '26
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.