Timeline for Circular, or missing, definition in set theory?
Current License: CC BY-SA 4.0
17 events
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| May 24, 2018 at 18:20 | comment | added | Burak | @FrankQuinn: ...some intuitive meaning to your symbols. But the point is, as Noah emphasized in his edit, whatever axiomatic system will be your "background" system, you have to either take a formalist approach or be willing to let go off your search for precise formulation of its semantics. | |
| May 24, 2018 at 18:17 | comment | added | Burak | @FrankQuinn: Contrary to the popular belief, mathematics cannot bootstrap itself. You cannot prove something without assuming anything. If you want to talk about models of a theory mathematically, then this discussion has to take place within some other axiomatic system. (This is the theory/metatheory distinction.) Clearly, whatever axiomatic system you start with cannot be given proper semantics, unless you are willing to talk about it within some other system. At this point, you have to take this axiomatic system as is. You are just manipulating some strings, you may (or may not) give... | |
| May 24, 2018 at 18:13 | comment | added | Burak | @FrankQuinn: $\exists x \forall y \neg y \in x$. In other words, there is an object $x \in X$ such that for all $y \in X$ it is not the case that $y E x$. That is, if we interpret E as the membership relation, from the perspective of the model $(X,E)$, the object $x \in X$ is the empty set. Is $x$ the real empty set? Not necessarily. It behaves like the empty set within the model $(X,E)$. What does this have to do with what you are asking? I am trying to illustrate that there is no circularity with ZFC being able to talk about models of (the formalized theory) ZFC... | |
| May 24, 2018 at 18:08 | comment | added | Burak | @FrankQuinn: I believe, what you are trying to ask at the core, may have been addressed in my following answer to a closely related question: mathoverflow.net/questions/248965/…. More specifically, what you are calling an implementation is simply a model $(X,E)$ of a (formalized) first-order theory. X is a set together with a binary relation which has an interpretation, just like your function $E$. If the model $(X,E)$ is a model of ZFC, then, for example, it satisfies the sentence... | |
| May 24, 2018 at 15:24 | comment | added | Frank Quinn | @burak Perhaps the question should be: what are the properties of the "universe" in a specific implementation. The revised version may be clearer. | |
| May 24, 2018 at 6:45 | comment | added | Monroe Eskew | @Qfwfq-- rather shameless. | |
| May 24, 2018 at 4:54 | comment | added | მამუკა ჯიბლაძე | Are you sure we know what points are to Euclid? One colleague claims that Euclid wrote drawing instructions, and points for him are markings sufficiently small for unambiguously determining how to draw a line through any two of them. Accordingly, one might take the stance that we only need set theory as a substrate to do mathematics, and should only worry about unambiguity. Then sets would be not the objects of the universe but tools to express ("draw") what you want to say or learn about objects of the universe. | |
| May 22, 2018 at 18:46 | comment | added | Qfwfq | @Burak: yes, you're of course right that the answer doesn't "use" platonism. Take my downvote as a "political" statement against a philosophical view of mathematics that I find immature and to be eradicated. Also this has few to do with your answer, but... you get my point ;) | |
| May 22, 2018 at 17:28 | comment | added | Burak | @Qfwfq: Well, if you do not like the idea of the universe of sets, we may choose a universe sets and restrict quantification to the objects in this universe, (informally) define truth of sentences with respect to this restricted universe and etc. The core of this answer has nothing to do with mathematical Platonism, it is trying to illustrate how the binary relation symbol $\in$ gets its meaning once you choose a universe to interpret it in. | |
| May 22, 2018 at 17:26 | comment | added | Qfwfq | Regardless from the rest of the answer, for me "Let me take a Platonist approach to elaborate" deserves a -1. Sorry. | |
| May 22, 2018 at 16:20 | history | edited | Burak | CC BY-SA 4.0 |
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| May 22, 2018 at 13:59 | history | edited | Burak | CC BY-SA 4.0 |
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| May 22, 2018 at 13:52 | history | edited | Burak | CC BY-SA 4.0 |
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| May 22, 2018 at 13:45 | history | edited | Burak | CC BY-SA 4.0 |
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| May 22, 2018 at 13:43 | comment | added | Itai | "I admit that I don't fully understand what your problem is" - At first I 'understood' the problem, but then I realized the question is in fact not about set theory, but rather about logic. Set theory here is only an example. The deeper question (as I understand it now) is "What is the difference between symbols (of the language, as in logic) and functions/relations in any model (of the set of axioms etc.)" | |
| May 22, 2018 at 13:30 | history | edited | Burak | CC BY-SA 4.0 |
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| May 22, 2018 at 13:24 | history | answered | Burak | CC BY-SA 4.0 |