## Tagged Questions

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### Reference request: Minimal Axiomatizations of PA over (+,x,<=).

Many years ago, when I was still a high school student, I came up with a certain first-order axiomatization of PA over the signature (+, x, ≤). Out of nostalgia, I've decided t …
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### Even XOR Odd Infinities?

Modular Arithmetic (MA) has the same axioms as first order Peano Arithmetic (PA) except $\forall x (Sx \ne 0)$ is replaced with $\exists x(Sx = 0)$. (http://en.wikipedia.org/wiki/ …
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### Why Does Induction Prove Multiplication is Commutative?

Andrew Boucher's General Arithmetic (GA2) is a weak sub-theory of second order Peano Axioms (PA2). GA has second order induction and a single successor axiom: \forall x \forall …
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### ERA, PRA, PA, transfinite induction and equivalences

I'm quite sure I don't understand very well the links between proof theoretical ordinals of theories, the axioms of transfinite induction and the objects a theory can prove to exis …
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### Gödel, Escher, Bach: b is a power of 10. [closed]

I’d like to verify if my formula correctly expresses that a number is a power of $10$, using the $\sf{TNT}$ language provided by Hofstadter in his famous book Gödel, Escher, Ba …
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### Can FPA really prove its consistency?

I will ask the question first and then explain. QUESTION: FPA can prove its own consistency in the Godelian sense. But can it really prove its consistency? FPA is a multi-sorte …
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### Provability in Second-Order Arithmetic without the Successor Axiom

Consider second-order Peano Arithmetic Z2, i.e. the two-sorted first-order theory with induction and comprehension. Remove the assumption about the totality of the successor relat …
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### Is an ultrafinitist Hilbert’s program doomed?

Hilbert's program is popularly understood as an attempt to justify infinitary mathematics with a finitary consistency proof. Godel's Second Theorem is usually considered as showin …
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### Definitions for Oddness

In the thread Even Xor Odd Infinities I defined odd models of Modular Arithmetic (MA) as models satisfying the axioms of MA and two first order statements. http://mathoverflow.net/ …
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### First-order vs second-order provability

Modular Arithmetic (MA) has the same axioms as first order Peano Arithmetic (PA) except ∀x(Sx≠0) is replaced with ∃x(Sx=0). Let MA2 be the second-order variation, with second-order …
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### Models of the natural numbers in ultrapowers in the universe.

Our question arises from wondering about the systems of natural numbers in models ZFC + Con(ZFC) and ZFC + $\neg$Con(ZFC). In thinking of the systems of natural numbers of these m …
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### How would one even begin to try to prove that a simple number-theoretic statement is undecidable?

This question is closely related to this one: http://mathoverflow.net/questions/27755/knuths-intuition-that-goldbach-might-be-unprovable. It stems from my ignorance about non-stand …
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### parameters in arithmetic induction axiom schemas

The induction schema of Peano Arithmetic is standardly given as the universal closure of $\phi(0)\land \forall x (\phi(x)\rightarrow \phi(x+1)) \rightarrow \forall x\phi(x)$. Howe …
Although, beyond any doubts, $ZFC$ is by and large the predominantly accepted theory of sets, there have been a few attempt to establish some serious competitors in town. I just …
Consider Peano's axioms — in its first-order version and without addition and multiplication — with its single injective function $S$: $(\forall x) \neg Sx = 0$ \$\Big …