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Timeline for Circular, or missing, definition in set theory?

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May 24, 2018 at 21:51 comment added Alec Rhea @FrankQuinn The book "Foundations of Mathematics" by Kenneth Kunen may be of some assistance to you if your library has a copy. It is very up to date (published 2009) and explicitly discusses the roles of set theory, logic and model theory as they pertain to securing a foundation for mathematics without issue or paradox.
May 24, 2018 at 20:01 comment added Timothy Chow (cont'd) Furthermore, any "domain in which first-order logic works reliably" that one might propose is open to the same objection: What is your foundational basis for believing that first-order logic works reliably in that domain? Any answer to that question is no less "circular" than the standard approach.
May 24, 2018 at 19:57 comment added Timothy Chow @FrankQuinn : If you think that using set theory to do model theory is 'circular' in some illegitimate sense, then I think you're still suffering from a basic confusion. The circularity is only apparent and is not real. If you insist on doing model theory in a non-set-theoretical way, then one could shoehorn it into some other framework; e.g., you can do a lot of finite model theory using arithmetic. There are probably ways to develop model theory using type theory. But I'd argue that this would be pointless; you'd be bending over backwards to avoid something that doesn't need to be avoided.
S May 24, 2018 at 19:33 history suggested Phil Tosteson CC BY-SA 4.0
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May 24, 2018 at 19:28 review Suggested edits
S May 24, 2018 at 19:33
May 24, 2018 at 16:07 history edited Noah Schweber CC BY-SA 4.0
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May 24, 2018 at 15:54 comment added Noah Schweber Since my edit is quite long, let me state my view: there can be no semantic approach to mathematics which doesn't fall prey to either the need to make philosophical commitments (e.g. "sets exist") or circularity; however, mathematics can be developed on a purely formalist foundation, and this provides us with a method for ignoring the circularity while still doing "semantic" mathematics and not making any sort of Platonist commitment.
May 24, 2018 at 15:52 history edited Noah Schweber CC BY-SA 4.0
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May 24, 2018 at 15:51 comment added Noah Schweber @FrankQuinn "I want an interpretation for "model" that does not presuppose any amount of set theory" The point of talking about "commitments" is that I'm claiming that such a thing doesn't exist. I'm not sure, meanwhile, what you mean by a "domain in which first-order logic works reliably" - perhaps the idea of doing logic inside a topos, viewed as an alternative notion of "set," is on-topic for this? Regardless, see my edit.
May 24, 2018 at 15:49 comment added Frank Quinn @noah Try this: rather than "commitments we need to make sense of the semantics for first-order logic", I am asking about a description of domains in which first-order logic works reliably. My experience is that "commitments" and "sense" often have no logical force. Also, I want an interpretation for "model" that does not presuppose any amount of set theory, in order to avoid circularity.
May 24, 2018 at 15:26 comment added Noah Schweber @FrankQuinn If I understand correctly, your "implementation" is exactly a model in the sense of first-order logic, and indeed talking about models presupposes some amount of set theory. I don't understand your claim that this is not about the logic or metatheory; this is, as far as I can tell, exactly about the logic and metatheory, namely what commitments we need to make sense of the semantics for first-order logic.
May 24, 2018 at 15:21 comment added Frank Quinn @noah please look at the revised version; maybe it is clearer. The question is about usable implementations, not about the logic or metatheory.
May 23, 2018 at 18:15 comment added Noah Schweber @FrankQuinn Can you clarify if I've interpreted your question correctly in my most recent comment? Once I understand what you're asking better, I can edit this answer to be more relevant.
May 22, 2018 at 17:08 comment added Noah Schweber @FrankQuinn So the circularity you're worried about is that we use "set" to define the semantics for first-order logic, but the things we believe about sets are expressed by a first-order theory to begin with (namely ZFC); is that right?
May 22, 2018 at 16:31 comment added Zuhair Al-Johar I like this graphical interpretation. In reality I'm one of the people who thinks there are just graphs, and 'set', 'class' are just a abstractions about parts of graphs. So I think the real backbone of foundation would ultimately be a kind of Mereotopological graph theory. Anyhow
May 22, 2018 at 13:57 comment added Timothy Chow @FrankQuinn : You might want to read the first few pages of my article on forcing, which addresses this apparent circularity. timothychow.net/forcing.pdf Another way out of the apparent circularity is to begin by treating sentences of ZFC as meaningless syntactic strings, not as "saying" anything about sets. After your machine has mindlessly generated enough strings, you then accidentally notice that the structure of these strings strongly resembles ordinary mathematical discourse, so you do a search-and-replace to convert journal articles to ZFC strings.
May 22, 2018 at 13:37 comment added Burak Noah, I believe this answer might be more confusing for those naively believing that ZFC answers the question "what is a set". Sets are simply objects in the universe. @FrankQuinn: Once you set up your set theory, you notice that there is a way to formalize model theoretic notions within this system. Given a formal theory $T$, which is really a set whose elements encode strings, a model of $T$ is a(n actual) set $X$ together with... What Noah is saying is that, if we take a model of ZFC, the metatheoretic notion of a set and the internal notion of a set inside $X$ are different things.
May 22, 2018 at 11:28 comment added Zuhair Al-Johar @FrankQuinn, no the statement "$X$ is a set" is not internal to an implementation of the axioms, actually there is no predicate "set" that is defined within the usual set theories (except in $MK$ were we artificially define it as an element of a class). Actually the statement $X$ is a set is very broad it involves objects within the universe of discourse of set theories and may involve the universe of discourse itself. "..is a set" is a characterization, not a definition internal to the theory.
May 22, 2018 at 10:47 comment added Frank Quinn @noah I would like your answer very much, except that the circularity is right up front with the word "set" in "... a set $X$ with a binary relation ...". A statement like "$X$ is a set" is internal to an implementation of the axioms, and not available to describe an implementation. If you change this to ".. a collection-of-elements $X$ with a binary relation.." ("sets" are then defined in terms of the relation) then my question is: what is a "collection-of-elements"?
May 21, 2018 at 21:20 comment added Noah Schweber @მამუკაჯიბლაძე Arguably, but that has nothing to do with any claimed circularity here. The point is that that distinction happens outside the theory ZFC, even if it motivated the construction of ZFC.
May 21, 2018 at 21:18 comment added მამუკა ჯიბლაძე But some (most?) theories are introduced to try and provide an exhaustive characterization of one particular model, no? I mean, in most cases this is in fact impossible and there are other different models that the theory is not able to distinguish from the "real thing". But we still can try to approximate it better by strengthening the theory. And to do so we must have the "ultimate" model somehow present. So, does not it make sense to distinguish some class of "natural" models which have the property that $wEu$ in such a model implies that $w$ really is an element of $u$?
May 21, 2018 at 20:59 history answered Noah Schweber CC BY-SA 4.0
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