# Questions tagged [theories-of-arithmetic]

Theories of arithmetic in first-order logic, such as Peano arithmetic, second-order arithmetic, Heyting arithmetic, and their subsystems and extensions.

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### Lunar arithmetic to help the set $\left\{ \mathbf{+}, \mathbf{\cdot} \right \}$ functionally complete?

If the goal of lunar arithmetic (=lunatic) to develop a minimal set of operations that can perform all the arithmetic: From the typical equation in part 9, is this possible to combining members of the ...
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### Dedekind-Peano axioms, but numbers have at most one successor

One can consider a variant of the Dedekind-Peano axioms in which one replaces the assumption that every number has exactly one successor by the assumption that every number has at most one successor, ...
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### Does visible nonstandardness imply visible ill-foundedness?

For $X\subseteq\mathfrak{M}\models \mathsf{TA}$, say that $X$ is $\mathfrak{M}$-disruptive iff there is some formula $\varphi$ in the language of arithmetic + a new unary predicate symbol $U$ such ...
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### What is the smallest countable limit ordinal in which 'lost melodies' occur

The question is in the title. This question is in response to the following paragraph found at the end of Prof. Hamkins' answer to my MathOverflow question, Are ITTM's necessary to compute Turing's &...
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### Can this weakish system of arithmetic express multiplication for second-sort numbers?

Consider a 2-sorted first-order logic with equality (for first-sort entities). The first sort consists of numbers, the second sort (which will be capitalized) of unary functions. There is one constant,...
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### How similar can a model of $I\Delta_0$ be to the intersection of all of its definable cuts?

Let $M$ be a model of $I\Delta_0$. Recall that a definable cut is a definable (possibly with parameters) subset $I$ of $M$ that is non-empty, downwards closed, and closed under successor. If we ...
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### What can $I\Delta_0$ prove?

What combinatorial and number-theoretic propositions can $I\Delta_0$ prove? Obviously there are an infinitude of them, but what are some well known theorems that can be proved in $I\Delta_0$, if any?
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### Reverse mathematics of Noetherian rings over $\mathbb{Q}$

Take the Hilbert Basis Theorem over the rational numbers in this form in the language of Second Order Arithmetic:  For every $n\in N$ every ideal of the polynomial ring $\mathbb{Q}[x_1,\dots,x_n]$ is ...
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### Did Edward Nelson accept the incompleteness theorems?

Edward Nelson advocated weak versions of arithmetic (called predicative arithmetic) that couldn't prove the totality of exponentiation. Since his theory extends Robinson arithmetic, the incompleteness ...
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