Timeline for What does the axiom of replacement mean and why should I believe it?
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Nov 21, 2021 at 16:17 | comment | added | user21820 | @TimothyChow: I don't disagree that mathematicians unconsciously use something that looks like replacement, but I disagree with saying that they unconsciously use replacement. If everything they do ends up being of the form I described, then it just shows that bounded replacement is intuitive, not that full replacement is. | |
| Nov 21, 2021 at 16:13 | comment | added | Timothy Chow | @user21820 The question here isn't whether unbounded replacement is needed in ordinary mathematics. It's whether mathematicians "unconsciously" construct sets using replacement. I'd claim that when mathematicians "unconsciously" use replacement, they aren't thinking about whether the formula is bounded or not. That we can recast their argument after the fact using bounded formulae is interesting, but not directly germane to the question, unless maybe you think we should never believe anything beyond the weakest axioms we need. | |
| Nov 20, 2021 at 18:09 | comment | added | user21820 | Hello! I came across the FOM post and was trying to see if anyone on MO had anything to say about it, so I found your post. I don't agree with the claim here, because clearly the sets that ordinary mathematics uses does not go beyond bounded ZFC (i.e. with Specification and Replacement restricted to bounded formulae). In particular, we can have Skolem functions witnessing Pairing and Powerset, and set-builder notation { E : x∈S ∧ Q } where E is a term with only free variable x and Q is a bounded formula, and we would be stuck in bounded ZFC but be able to do all ordinary mathematics easily. | |
| Dec 16, 2019 at 3:21 | comment | added | Timothy Chow | @RémiPeyre : What you say is true. On the other hand, it's not clear to me what more convincing answer can possibly be given to "why should I believe this axiom" than "it seems self-evident and leads to no known contradictions." (Saying that we find ourselves using it naturally is another way of saying that most people find it self-evident.) Since we're asking about an axiom we can't ask for a proof that it is correct. If this kind of justification isn't good enough then it seems to me we should just quit asking for reasons to "believe" in axioms and become formalists. | |
| Nov 23, 2019 at 19:47 | comment | added | Rémi Peyre | This answer does not convince me: after all, the “fact” that there exists a set of all sets directly translates the definition of the universe of set theory, and precisely reflects our intuition; and yet it leads to a contradiction… So, I do not think that “we tend to use the replacement axioms it spontaneously” is a good reason for believing in them! | |
| Jan 13, 2016 at 17:39 | comment | added | Johannes Hahn | @AsafKaragila You'd need permit A38 for that. | |
| Jan 12, 2016 at 19:06 | comment | added | Asaf Karagila♦ | @Johannes: Bring me the forms I need to fill to have you congratulated for this reference to Futurama! | |
| Jan 12, 2016 at 18:07 | comment | added | Emil Jeřábek | cs.nyu.edu/pipermail/fom/2007-September/011918.html | |
| Jan 12, 2016 at 18:01 | comment | added | Johannes Hahn | Technically correct - The best kind of correct ;-) | |
| Jan 12, 2016 at 10:39 | comment | added | Emil Jeřábek | Hmm. In the standard von Neumann representation of natural numbers, $P(\mathbb N)\subseteq P(P(\mathbb N))$, so it is technically correct, but weird. | |
| Jan 12, 2016 at 4:40 | comment | added | Timothy Chow | @StevenLandsburg: Yes, I think so, but I'll leave it as it is because it's a quotation. | |
| Jan 12, 2016 at 1:16 | comment | added | Steven Landsburg | Should that iterated power set be just a power set? | |
| Jan 12, 2016 at 0:35 | history | answered | Timothy Chow | CC BY-SA 3.0 |