# Tagged Questions

Questions about mathematical logic, including model theory, proof theory, computability theory (a.k.a. recursion theory), and non-standard logics. Questions which merely seek to apply logical or formal reasoning to other areas of mathematics should not use this tag. Consider using one of the ...

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### Model of Robinson Arithmetic but not Peano Arithmetic

I am curious how can structure of with linear successor function that do not admin induction looks like. In other words I would like to see structure of Peano Arithmetic without induction. I believe ...
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### Beginner material for mathematical logic

I am looking for study and beginner material to study mathematical logic. I understand that it is a very broad topic but I would like to know what the best path there is to learning mathematical logic....
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### Quantification over the set(?) of predicates

When learning set theory and logic, one fact that popped up a handful of times was that we did not quantify over predicates. To quote the notes I took in class on the axiom of unrestricted ...
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### A terminology to analysts

When analysts say "$\epsilon$ (or whatever greek symbol) can be chosen arbitrary small", do they really just mean we can take $\epsilon = 0$ or $\epsilon \to 0$ later? When I asked myself this ...
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### Defining new symbols (abbreviations) in first-order logic

In first order logic it is common (and just about necessary) to introduce new symbols which have been defined in terms of the "fundamental" symbols of a given theory. For instance, the signature of ...
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### The relation between logic and algebra.

What's the relation between logic and algebra? Can one be thought of as a special case of the other?
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### Models of real numbers combined with Peano axioms

Suppose you take the axioms for a Dedekind-complete ordered field and weaken the Dedekind-completeness axiom to the corresponding weaker first-order axiom schema (e.g. replace the left and right sets ...
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### Justification of ZFC without using Con(ZFC)?

I saw this post by Carl Mummert, which describes how one can 'see' the reasonableness of ZFC, via iterating the power-set operation starting from the empty set. However, this iterations proceed in ...
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### substituting a variable in a formula (in logic)

What kind of mathematical object is this substitution(is it a function or what). We assuming set of variables exist.
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### Prove this proposition concerning a theory with ∀∃-axiomatization part II.

This is a continuation of a problem I asked yesterday seen here: Prove this proposition concerning a theory with ∀∃-axiomatization The setting is reproduced below for easy reference. Setting A ...
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### What does a condition being sufficient as well as necessary indicates?

I have a question in a book I am solving(Discrete Structures by Kolman, Busby & Ross). I am unable to make sense from the question. It is stated below, Show that k is odd is a necessary and ...
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### $A⇒(B \lor C)$ and $[(A \Rightarrow B) \lor (A \Rightarrow C)]$

[(A⇒ B∨C)] ⇒ [A⇒(¬B⇒C)] ⇒[(A⇒¬B)⇒(A⇒C)] ⇒ [¬(A⇒¬B)∨(A⇒C)]⇒[(A∧B)∨(A⇒C)] [(A⇒B)∨(A⇒C)] is equivalent to A⇒(B∨C). Can I prove [(A∧B)∨(A⇒C)] ⇒ [A⇒(B V C)]? or is there problem in the proof above ...
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### What is definability in First-Order Logic?

Can someone explain to me the definition of definability in first-order logic in simple terms and with an example? I would appreciate this. I just want to really understand this. Thank you. Here is ...
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### Are the real numbers really uncountable?

Consider the following statement Every real number must have a definition in order to be discussed. What this statement doesn't specify is how that loose-specific that definition is. Some examples ...
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### Recommendations for Intermediate Level Logics/Set Theory Books

I currently don't know what books I should read after having studied elementary logics/set theory, so I'd be glad if I can get some recommendations. My classes used the following two books for logics/...
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### Example of a proof using the axiom of commensurability

I'm teaching our intro to proofs course (well, one of them) and one of the classic illustrations of an overturned "axiom" is the Greek axiom of commensurability, which stated in geometric terms the ...
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### $F[t]$ has undecidable positive existential theory in the language $\{+, \cdot , 0, 1, t\}$

Consider the ring $F[t, t^{-1}]$ (the polynomials in $t$ and $t^{-1}$ with coefficients in the field $F$). Theorem 1. Assume that the characteristic of $F$ is zero. Then the existential theory ...
### Every elementary submodel of $H(\aleph_1)$ is transitive
I am trying to solve the first part of Exercise II.17.30 in Kunen's Foundations of Mathematics which asks: Prove that every $A \preccurlyeq H(\aleph_1)$ is transitive. Here we are working over ...