**50**

votes

**3**answers

6k views

### Nelson's program to show inconsistency of ZF

At the end of the paper Division by three by Peter G. Doyle and John H. Conway, the authors say:
Not that we believe there really are any such things as infinite sets, or that the Zermelo-Fraenkel ...

**35**

votes

**7**answers

2k views

### 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: Knuth's intuition that Goldbach might be unprovable. It stems from my ignorance about non-standard models of arithmetic. In a comment on the other ...

**33**

votes

**2**answers

2k views

### What arithmetic information is contained in the algebraic K-theory of the integers

I'm always looking for applications of homotopy theory to other fields, mostly as a way to make my talks more interesting or to motivate the field to non-specialists. It seems like most talks about ...

**29**

votes

**3**answers

2k views

### Why do stacked quantifiers in PA correspond to ordinals up to $\epsilon_0$?

I am trying to understand why induction up to exactly $\epsilon_0$ is necessary to prove the cut-elimination theorem for first-order Peano Arithmetic; or, as I understand, equivalently, why the length ...

**27**

votes

**3**answers

1k views

### 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 $\exists x(Sx = 0)$.
...

**26**

votes

**8**answers

3k views

### Arithmetic fixed point theorem

I want to understand the idea of the proof of the artihmetic fixed point theorem. The theorem is crucial in the proof of Gödel's first Incompletness theorem.
First some notation: We work in $NT$, the ...

**26**

votes

**1**answer

413 views

### Decidability of equality of expressions built using 1,+,-,*,/,^

Consider expressions built using number $1$, arithmetical operators $+, -, *, /$ and exponentiation ^ (in case of multiple values, the principal value is assumed, the same way as it implemented in ...

**24**

votes

**3**answers

2k views

### Composite pairs of the form n!-1 and n!+1

It's well known that the numbers of the form $n!\pm1$ are not always prime. Indeed, Wilson's Theorem guarantees that $(p-2)!-1$ and $(p-1)!+1$ are composite for every prime number $p > 5$.
Is ...

**15**

votes

**4**answers

2k views

### What is induction up to epsilon_0?

This is a question asked out of curiosity, and because I can't understand the wikipedia page.
I have often been told that PA cannot prove the validity of induction up to $\epsilon\_0$, which has been ...

**13**

votes

**6**answers

1k views

### Non-constructive proofs of decidability?

Are there examples of sets of natural numbers that are proven to be decidable but by non-constructive proofs only?

**12**

votes

**2**answers

1k views

### Is there a known natural model of Peano Arithmetic where Goodstein's theorem fails?

(I've previously asked this question on the sister site here, but got no responses).
Goodstein's Theorem is the statement that every Goodstein sequence eventually hits 0. It turns out that this ...

**12**

votes

**1**answer

240 views

### What metatheory proves $\mathsf{ACA}_0$ conservative over PA?

Simpson's book shows $\mathsf{ACA}_0$ is conservative over $\mathsf{PA}$ in the natural way by model theory using definable subsets. Of course, $\mathsf{ACA}_0$ being conservative over PA is ...

**12**

votes

**1**answer

588 views

### 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-sorted first-order ...

**12**

votes

**1**answer

371 views

### Transfinitely extending $\sf PA$ — can we get stronger than $\sf ZFC$?

Let $\sf PA$ denote the theory of natural numbers with constants $(0, 1)$ and binary operators $(+,\times)$ based on the first-order predicate calculus with equality, having the following axioms, ...

**11**

votes

**1**answer

392 views

### Does any lower bound on proofs of FLT improve Shepherdson 1965?

In 1965 Shepherdson proved that FLT is independent of the fragment of PA that uses only open induction and signature $0,S,+\times$. Indeed $2x+1\neq 2y$ is independent of that fragment. Schmerl ...

**10**

votes

**5**answers

1k views

### Alternative Arithmetics

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 quote two of them ...

**10**

votes

**3**answers

725 views

### Reducing ACA₀ proof to First Order PA

According to the Wikipedia ACA0 is a conservative extension of First Order logic + PA.
http://en.wikipedia.org/wiki/Reverse_Mathematics
First of all I have a few questions about the proof:
a - What ...

**10**

votes

**2**answers

625 views

### Is Robinson Arithmetic biinterpretable with some theory in LST?

Let ZFC$^{\text{fin}}$ be ZFC minus the axiom of infinity plus the negation of the axiom of infinity. It is well-known that ZFC$^{\text{fin}}$ is biinterpretable with Peano Arithmetic. In this sense ...

**9**

votes

**4**answers

3k views

### Historically first uses of mathematical induction

I'm interested in find out what were some of the first uses of mathematical induction in the literature.
I am aware that in order to define addition and multiplication axiomatically, mathematical ...

**9**

votes

**3**answers

253 views

### Uncountable model of bounded arithmetic with an elementary end extension

Theorem 1.53 (3) in page 227 of Hajek and Pudlak's book, Metamathematics of First-Order Arithmetic, says:
Theorem. If $M$ is a countable model of $I\Delta_{0}$ such that $M$ has a proper elementary ...

**9**

votes

**2**answers

385 views

### Z_2 versus second-order PA

These days, Peano Arithmetic ($PA$) refers to the first-order version of the axioms, where induction is only over formulas referring to natural number variables. Peano's original version of the ...

**9**

votes

**1**answer

663 views

### 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 showing this is not ...

**8**

votes

**9**answers

3k views

### Is PA consistent? do we know it?

1) (By Goedel's) One can not prove, in PA, a formula that can be interpreted to express the consistency of PA. (Hopefully I said it right. Specialists correct me, please).
2) There are proofs ...

**8**

votes

**3**answers

294 views

### If an oracle Turing machine halts with every infinite arithmetic oracle, can it fail to halt with some non-arithmetic oracle?

Let $e$ be an index of an oracle Turing machine program and $k$ be some natural number. Let us say that a subset of $\mathbb N$ is arithmetic if it is definable in the model $\langle \mathbb ...

**8**

votes

**1**answer

312 views

### The (un)decidability of Robinson-Arithmetic-without-Multiplication?

I asked this over at math.stackexchange, and though a number of people were interested enough to vote up the question, I didn't get an answer -- which makes me wonder whether it isn't quite so ...

**8**

votes

**1**answer

355 views

### Does higher order arithmetic interpret the axiom of choice?

By second order arithmetic I mean the axiomatic theory $Z_2$, that is Peano arithmetic extended by second order variables with the full comprehension axiom, and not defined semantically using power ...

**8**

votes

**2**answers

750 views

### The different Branches of Arithmetics

... "and then the
different branches of Arithmetic--
Ambition, Distraction, Uglification,
and Derision."
(Alice in Wonderland, chapter IX: the Mock Turtle's story)
As a child I ...

**7**

votes

**4**answers

1k views

### Incompleteness and nonstandard models of arithmetic

The following are a collection of doubts, some of which shall have concrete answers while others may have not. Any kind of help will be welcome.
Reading Peter Smith's "Gödel Without (Too Many) ...

**7**

votes

**3**answers

1k views

### Gödel's Incompleteness Theorem and the complexity of arithmetic

In How complicated can structures be? Jouko Väänänen says:
The guiding result of mathematical logic is the Incompleteness Theorem of Gödel,
which says that the logical structure of number theory ...

**7**

votes

**3**answers

404 views

### Elementary end extensions of models of Peano Arithmetic in uncountable languages

A well-known theorem of Mills asserts that there is a model of Peano Arithmetic $M$ in an uncountable language such that $M$ has no elementary end extension (e.e.e.). I ask whether every complete ...

**7**

votes

**4**answers

555 views

### Reference Request: Non-Standard Models of PA

I am attempting to write an expository paper on non-standard models of PA that is accesible to students taking an introductory graduate course in mathematical logic (covering Godel's incompleteness ...

**7**

votes

**1**answer

956 views

### models of PA which are isomorphic but not elementarily equivalent?

On page 164 of his book Models of Peano Arithmetic, Kaye states Friedman's Theorem:
Let $M{\vDash}PA$ be nonstandard and countable, let $a\in M$ and let $n\in {\mathbb N}$. Then there is a proper ...

**7**

votes

**1**answer

603 views

### Has Ribet's theorem been proved using only finite powers of primes?

Ribet proved the Serre epsilon conjecture using $p$-adic Galois representations (http://math.berkeley.edu/~ribet/Articles/invent_100.pdf). Can someone show how to replace all use of $p$-adics in ...

**7**

votes

**1**answer

248 views

### The definition of < in Robinson's Q

I recently had to explain how the basic axioms in Simpson's Subsystems of Second Order Arithmetic were interpretable in Robinson's Q. Most of the axioms are actually the same, except that Simpson ...

**7**

votes

**1**answer

267 views

### Forcing, cuts, and Dedekind-finite cardinalities

Tl;dr version: there are two natural classes of cuts in the nonstandard model of arithmetic consisting of the Dedekind-finite sets (if, in fact, they constitute such a model); both these classes are ...

**7**

votes

**1**answer

493 views

### Nelson natural number objects in a topos (say)

Nelson's predicative arithmetic (survey article) is a very weak system of arithmetic extending Robinson's $Q$ (Wikipedia).
We can have natural number objects in a topos, or even a merely finitely ...

**6**

votes

**1**answer

159 views

### A well-behaved $A$ that is almost contained in every element of some filter for a countable arithmetically closed family $\mathfrak X$

The question has relevance for constructing Scott sets with certain extra desirable properties.
Suppose that $\mathfrak X$ is a countable arithmetically closed family of subsets of $\mathbb N$: ...

**6**

votes

**1**answer

237 views

### Does the totality of Ackermann's function prove the consistency of $\Sigma_1$-induction?

It is well known that Ackermann's function is not primitive recursive. Therefore, the theories of primitive recursive arithmetic (PRA) and of $\Sigma_1$-induction ($I\Sigma_1$) cannot prove the ...

**6**

votes

**1**answer

482 views

### How to locate the paper that established Robinson Arithmetic?

If I'm not mistaken, it was in his seminal paper “An Essentially Undecidable Axiom System”, published in
Proceedings of the International Congress of Mathematics (1950), 729–730,
where R.M. ...

**5**

votes

**4**answers

483 views

### parameters in arithmetic induction axiom schemas

The induction schema of Peano Arithmetic is standardly given as the universal closure of $\phi(0)\land \forall x (\phi(x)\rightarrow \phi(x+1)) \rightarrow \forall x\phi(x)$. However, since the ...

**5**

votes

**3**answers

2k views

### Non-computable but easily described arithmetical functions

I have read about the existence of functions of the kind described in the title in several places, but never seen an instance of them. Sorry if this is too much an elementary question to be posted ...

**5**

votes

**2**answers

866 views

### Set Theory inside Arithmetics via the Ackermann Yoga

Among the basic results of logic which, simple as they are, never fail to intrigue me, is Ackermann's interpretation of $ZF$-Infinity in $PA$ (see for refs this MO question and here for an excellent ...

**5**

votes

**2**answers

550 views

### Weakest subsystems of second order arithmetic for mathematical logic

It is known that to prove completeness of first-order logic for countable languages WKL0 is enough. But, is it the weakest subsystem where one can prove it?
What about the incompleteness theorems? Is ...

**5**

votes

**2**answers

355 views

### 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 induction.
...

**5**

votes

**2**answers

841 views

### A question about open induction

An old theorem of A. J. Wilkie (Some results and problems on weak systems of arithmetic, Logic Colloquium '77) asserts that a discretely ordered ring $R$ can be extended to a model of open induction ...

**5**

votes

**2**answers

149 views

### Models of PRA/EFA with induction on $X$ but not $\omega^X$

As I currently understand it, induction on formulas containing $N+1$ first-order quantifiers is required to prove the well-ordering of the ordinal $(\omega \uparrow\uparrow N) < \epsilon_0$, that ...

**4**

votes

**7**answers

2k views

### Are real numbers countable in constructive mathematics?

We are talking about ordinary reals in constructive mathematics.
Let represent each real number by infinite converging series:
$$r = [\;(a_0,b_0),(a_1,b_1),...,(a_i,b_i),...\;]$$
$$where\quad a_i ...

**4**

votes

**2**answers

807 views

### Axiom to exclude nonstandard natural numbers

In Peano Arithmetic, the induction axiom states that there is no proper subset of the natural numbers that contains 0 and is closed under the successor function. This is intended to rule out the ...

**4**

votes

**2**answers

240 views

### Goedelizability and decidability of a property of Peano formulas

Sorry for not knowing the answers to these elementary questions:
Is the property of formulas of the first-order language of Peano arithmetic of "defining a finite set of natural numbers" ...

**4**

votes

**3**answers

463 views

### 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 relationship (the ...