All Questions
Tagged with peano-arithmetic or theories-of-arithmetic
73 questions with no upvoted or accepted answers
2
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75
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Can all the strongly provable theorems of $\sf PA+\neg Con(PA)$ be captured in an effective manner through alternative kind of provability?
If we extend $\sf PA$ with the following axiom asserting its own inconsistency:
Inconsistency: $\exists x: \operatorname {Proof}_{\sf PA} (x, \ulcorner 0=1 \urcorner)$
For short denote this axiom by $\...
- 11.3k
2
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0
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192
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Can PA be acyclically complete?
Any formula $\phi$ in the first order language of arithmetic is to be called acyclic if and only if we can associate with it an acyclic undirected graph whose nodes are the variable symbols occurring ...
- 11.3k
2
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0
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79
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Which sets of natural numbers are "lambda-analytic"?
Begin with a bit of notation. Let $t = t_0, \ldots, t_d$ be a finite sequence of real numbers. Define
$$\lambda^t(x) = x^{t_0} \log(x)^{t_1} \log(\log(x))^{t_2} \cdots.$$
for all real numbers $x \in ...
- 13.3k
2
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0
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137
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Can we extend the projectively extended real line with a single number that stands for division of zero by zero?
If we work within $\hat{\mathbb R} = \mathbb R \cup \{\infty\}$, i.e. one point compactification of the real line.
We extend $<$ relation on $\mathbb R$ to $\hat <$ defined as:
$ x \ \hat{<} \...
- 11.3k
2
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0
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237
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Representing iteration of a function in PA
Let $\mathscr{L}$ be a (recursive) FOL language, with numeral symbols $\underline{0},\underline{1},\ldots$. Let $T$ be a recursive, consistent theory, containing PA (or even just Robinson arithmetic)....
- 18.7k
2
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0
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84
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Seeking name for an order raising operator in Higher Order Arithmetic.
Any class $X$ of order $j$ in HOA is in bijection with the order $j+1$ class built up from singletons $\{x\}$ of natural numbers $x$ just the way that $X$ is built up from the numbers $x$. And of ...
- 11.1k
2
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223
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Is this fragment of arithmetic on $p^{-\infty} {\mathbb Z}$ decidable?
Let $p$ be a prime number. Consider the abelian group $p^{-\infty} {\mathbb Z} = \bigcup p^{-n} {\mathbb Z}$ consisting of rational numbers whose denominator is a power of $p$, under addition.
View $...
- 13.3k
1
vote
0
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90
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About synonymy relationships around these two theories?
The following question is about patterns of synonymy relationships around two theories, $T^+$ and $\sf PA$.
For purposes of self inclusiveness I'll re-iterate $T$ and its extensions.
$\textbf{Logic:}$ ...
- 11.3k
1
vote
0
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129
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Is set theory interpretable in infinite primitive recursive arithmetic?
In A Formalization of the Theory of Ordinal Numbers, Takeuti interprets $\sf ZFC$ in a first order theory extending first order arithmetic to the infinite ordinal realm, while at the same time ...
- 11.3k
1
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118
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Which real functions benefit from the Fundamental Theorem of Interval Analysis?
I'm reading Introduction to Interval Analysis, by Moore, Baker & Cloud and complementing it with Global Optimization using Interval Analysis, by Hansen & Walster.
Theorem 5.1 - Fundamental ...
1
vote
0
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117
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Can this type theory interpret second order arithmetic?
Language: multi-sorted first order logic with equality and membership, where for each natural $t$ there is a set $x^t$ of sort $t$. Equality "$=$" only occurs between variables of the same ...
- 11.3k
1
vote
0
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148
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Why doesn't $\mathsf{B}\Sigma_2$ hold in $\mathsf{RCA}_0$?
For a formula $\varphi(i,u)$ of arithmetic, the bounding principle for $\varphi$ is the statement
$$\forall m \, \Big( \big( \forall i<m\ \exists u\ \varphi(i,u) \big) \to \big( \exists v\ \forall ...
- 798
1
vote
0
answers
107
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Formalization in PA in the Kritchman-Raz proof
In their paper Kritchman and Raz present a proof of Gödel's second theorem using Kolmogorov complexity. To make it work, they operate in some (weak) formal theory $T$ that incorporates some arithmetic,...
- 189
1
vote
0
answers
194
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Induction on open formulas vs. Induction on $\Pi_1$ formulas
There are infinitely many extension to Robinson's $Q$ arithmetic many of which are defined by adding an axiom schema of induction for particular set of formulas.
I am confused about the theory $\text{...
- 173
1
vote
0
answers
346
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approaching the border between absolute convergence and divergence of series
Let us consider absolute convergent series $\ell^{1^+}$ ordered under eventual dominance (mod finite) $<^*$. T. Bartoszynski proved that unbounded number ${\frak b}(\ell^{1^+}, <^*)$ equals ...
- 88
1
vote
0
answers
205
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Lowest Turing degree that allows a Turing machine to tell whether $\operatorname{Con}(PA)$?
Let $T$ be a given turing machine. We say that $T$ decides $\operatorname{Con}(PA)$ if $PA + \operatorname{Con}(PA) \vdash T \text { accepts}$ and $PA + \lnot \operatorname{Con}(PA) \vdash T \text { ...
- 6,403
1
vote
0
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113
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How do I justify these nontheorems in the absence of the Existence Property for $PA$
Let $\Pi$ be the provability predicate for $PA$. I want to conclude that $PA\nvdash\exists x(\alpha(x)\wedge\lnot\Pi\ulcorner \alpha(\overset{.}{x}) \urcorner)$ and $PA\nvdash\exists x(\lnot \alpha(x)\...
- 3,574
0
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0
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73
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Least number principle for IOpen fragment of Peano Arithmetic
Is it possible to prove the least number principle in IOpen fragment of Peano Arithmetic, i.e. having induction only for Open formulas?
- 19
0
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0
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152
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What is the strength of allowing multiple predecessor numbers?
If we have a theory of numbers, pairs of numbers, and sets of those, and axiomatize that the relation $<$ on numbers is both extensional and well founded, then this theory would prove all PA axioms ...
- 11.3k
0
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0
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104
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Multivariate polynomial with infinite but discrete roots on one variable
I want to know if there exists a polynomial $ P(z, x_1,x_2,\ldots,x_n)$ over the rationals such that the set
$$
Z_P = \{z | \exists x_1,\ldots,x_n. P(z, x_1,x_2,\ldots,x_n) = 0 \} \subsetneq \mathbb Q
...
- 163
0
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0
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315
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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. Even XOR Odd Infinities?
$O1) \forall x(x=0 ...
- 687
-1
votes
2
answers
638
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Peano axioms— mathematical induction and other axioms
The Peano axioms of $\Bbb N$ are:
$1 \in \Bbb N$, i.e. $\Bbb N$ is not empty and contains an element denoted by $1$.
Every natural number has a successor, i.e. $\forall n\in\Bbb N, \exists!s(n)\in\...
- 101
-4
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0
answers
135
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Which arithmetic\set theory is synonymous with this theory?
$\textbf{Logic:}$ Mono-sorted first order logic with equality.
$\textbf{Extralogical Primitives: } <, \in$
Define: $x > y \iff y < x$
Define: $x \leq y \iff x < y \lor x=y$
$ \textbf{...
- 11.3k