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### How to define Set Theory and First Order Logic without circular reference? [duplicate]

Axiomatic set theory is basically a first order logic theory, so I believe we must certainly begin with a treatment of first order logic. However, to define a first order logic vocabulary we need to ...
42 views

### Sets, logic, and formal languages. Which precedes the other? [duplicate]

Recently I was reading about relations, and one passage stated "The notion of logical equivalence is, as its name suggests, an equivalence relation on the set of propositional terms". Now, to me ...
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### Question about set theory and first order logic [duplicate]

I was studying first order logic by a book and the author uses set theory to define some concepts like models, structures, etc. Everything fine here, but then i started to read about ZFC set theory ...
7k views

### Does mathematics become circular at the bottom? What is at the bottom of mathematics? [duplicate]

I am trying to understand what mathematics is really built up of. I thought mathematical logic was the foundation of everything. But from reading a book in mathematical logic, they use "="(equals-sign)...
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### An (apparently) vicious circle in logic

Can someone please help me with this following exercise 4.4 (p. 114) from the Mathematical Logic book of Ebbinghaus et al(this is not homework, but rather something that has been bugging me for a long ...
3k views

### Building the integers from scratch (and multiplying negative numbers)

Now I understand that what I am about to ask may seem like an incredibly simple question, but I like to try and understand math (especially something as fundamental as this) at the deepest level ...
246 views

### There is a book of mathematical logic that doesn't use informal arguments to build it theory?

I was taking a look to many textbooks of mathematical logic and I find them very unpleasant to read due to it lack of formalism and rigor at the metalanguage level. Let me explain: in all these ...
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### Function that have values that we can't determine (in principle - not because they are to difficult to compute)

I know that in some cases one has to exhibit functions like $f\equiv1$ if some famous conjecture is true and $f\equiv0$ else. With this I don't have a problem, because I perceive this function as ...
94 views

### set theory - sets containing something else

In a book about logic (propositional logic), author uses sets to describe propositional variables, operators (eg. $A=\lbrace \neg,\wedge,\vee,\Rightarrow,\Leftrightarrow\rbrace$)... How can those sets ...
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### First-Order Languages and Circular Reasoning

I'm reading a book on Mathematical Logic (on my own) and from the beginning there are terms such as "functions" and "relations", but the only definitions of these words that I know are in terms of ...
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### Trying to understand the difference between metatheory and theory and circularity

First off I just want to say that I understand that a model is not the same as the thing it models. I've already read several answers on this topic so I am looking for a new answer to hopefully ...
113 views

### What is the dependency hierarchy in foundational mathematics?

I am confused how different frameworks build on each other, or if there are any frameworks that are circular and we just sort of handwave it away. What is the order of what builds on what in terms of:...
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### How does ZFC solve the problem of alphabet in formal languages?

(In case someone thinks this is another question about the seeming circularity in formal languages and is going to downvote because of this, it's really not; don't downvote yet, keep reading) Perhaps ...