Questions tagged [peano-axioms]
For questions on Peano axioms, a set of axioms for the natural numbers.
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Peano axioms - do we need a specific property to show that the principle of mathematical induction implies the "correct" set of natural numbers?
From Terence Tao's Analysis I, Axiom 2.5 for the natural numbers reads
My intuition behind this axiom is that every natural number is an element of a "chain" of natural numbers that goes ...
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Formally how do we view finite sets
This might be silly, but I have been thinking about how we would work with finite sets very formally.
So, $\{1,2,3,\cdots,n\} = \{k \in \mathbb{Z}^+ \mid k \leq n\}$ gives a representee for which any ...
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Confusion about Löb's theorem [duplicate]
To quote wikipedia:
Löb's theorem states that in any formal system that includes PA, for any formula P, if it is provable in PA that "if P is provable in PA then P is true", then P is ...
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Is pointwise definability of a model of PA equivalent to it being the standard model? [duplicate]
The standard model of Peano Arithmetic is pointwise definable, because every finite natural number is parameter-free definable. What about the converse? That is, if a model $M$ of PA is pointwise ...
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What are the parameter-free definable elements of a model of Peano Arithemetic?
Let $M$ be a model of Peano Arithmetic. What are the parameter-free definable elements of $M$? I conjecture that they are precisely the standard natural numbers, meaning, no nonstandard infinite ...
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Non recursive provably total functions
It is provable that every primitive recursive function is total in 1st order PA. Some non primitive recursive functions are also provably total in PA.
Can we show that a function is totally provable ...
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Meaning of Provable recursiveness
Is there any difference between provably total function and provable recursiveness of a function in first order PA ?
From provably total I mean that the totality of the function itself is provable in ...
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Peano axioms-Analysis I by Terence Tao
Statement: Terence Tao in his book Analysis I states that the set N = {0,0.5,1,1.5,2,...} satisfies peano axioms 1 to 4.
Axiom 2: if n is a natural number, n++ is also a natural number
Definition 2.1....
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Is "standard $\mathbb{N}$" in fact not "fully formalizable"?
Note: "Update" at the end of this question hopefully summarizes/clarifies the original language (original text left in place for context).
Philosophical Preface: For the purposes of this ...
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Proving the set of true expressions in a theory cannot be expressed in this theory
Suppose we have a first-order theory $T_C$ that includes a binary function $C$.
$C$ is a bijection $\Sigma \to \mathbb{N}$ where $\Sigma$ is the alphabet of $T_C$.
The function $C$ is defined as ...
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Is a set $T$ containing all natural numbers identical to $\mathbb{N}$?
In this comment, I see that Mauro Allegranza and I were saying the same thing as shown in the table below:
Allegranza
Zeynel
$0 \in T$
$0 \in T$
$n \in T$
$n \in T$
$S(n) \in T$
$S(n) \in T$
$ T =...
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How to find what a definition defines?
Are these definitions of the "+" and "mod" operators?
$m+0=m$........(1)
$0 \;\text{mod} \;2 = 0$.....(2)
To me, (1) defines the identity property of zero and (2) defines zero as ...
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There are no loops in a Peano system. I've attempted a proof. Is my proof correct?
Definition (Peano systems). Suppose $P$ is a set, $1 \in P$, and $S: P \to P$ is a function. The triple $(P, S, 1)$ is a Peano system if the following conditions hold.
(P1) $\forall x (1 \neq S(x))$, ...
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How to prove commutative from Peano axioms WITHOUT proving associativity first?
I am reading this https://en.wikipedia.org/wiki/Peano_axioms#:~:text=The%20Peano%20axioms%20define%20the,0%20is%20a%20natural%20number.
They use S(x) for successor ...
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Is Russell's proof of addition with Peano's 5. Axiom valid?
This is a follow up question to my previous question: Why define addition with successor?
In this one I'd like to ask about Russell's use of Peano's 5. Axiom to prove his definition of addition:
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