Linked Questions

8
votes
0answers
189 views

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 ...
0
votes
0answers
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 ...
1
vote
0answers
37 views

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 ...
67
votes
8answers
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)...
20
votes
4answers
2k views

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 ...
10
votes
3answers
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 ...
3
votes
2answers
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 ...
2
votes
2answers
121 views

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 ...
3
votes
2answers
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 ...
1
vote
1answer
252 views

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 ...
1
vote
1answer
93 views

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 ...
2
votes
1answer
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:...
2
votes
1answer
108 views

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 ...
1
vote
0answers
217 views

Are there textbooks on logic where the references to set theory appear only after the construction of set theory?

In textbooks on logic the authors usually use the notions of set and map immediately, long before the set theory is constructed. That is strange for me, and I want to ask if anybody can advise me a ...
3
votes
0answers
200 views

How should one understand the foundation of set theory?

I have read the answer of Carl Mummert for the question on how to avoid circularity. I would like to ask further as I want to study models of set theory. As I understand, with say assuming the ...

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