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Tagged with peano-axioms proof-explanation
20
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Using Peano's axioms to disprove the existence of self-looping tendencies in natural numbers
Let me clarify by what I mean by "self-looping". So, we know that Peano's axioms use primitive terms like zero, natural number and the successor operation. Now, I want to prove that the ...
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Infinite statements from finite axioms
I want to know if a given finite subset of axioms of PA1 ( 1st order peano arithmetic ) can prove infinite sentences in PA1 such that those proofs need no other axioms except those in the given finite ...
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is arithmetic finitely consistent? [duplicate]
Let's take PA1( First order axioms of peano arithmetic ) for example. From godel's 2nd incompleteness theorem, PA1 can't prove its own consistency, more specifically it can't prove that the largest ...
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Proof of $\forall y(Sx + y = S(x + y))$ in Peano Arithmetic.
I am following a proof of $\forall y(Sx + y = S(x + y))$. The base case is $Sx + 0 = S(x + 0)$, and for its proof the author says this:
"If $y = 0$, note that by PA2 ($\forall x x + 0 = x$) we ...
- 496
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2
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Proof in Peano Arithmetic
I am trying to prove $S0 \times SS0$ using the axioms of Peano Arithmetic. The axioms are:
$\forall x \hspace{0.05cm}0 \ne Sx$
$\forall x \forall y \hspace{0.05cm} (Sx = Sy \rightarrow x = y)$
$\...
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Peano' s Systems
Lets consider $(A,g,a_{0})$ , where $A$ is a set, $g$ is a function and $a_{0}$ is the first element of $A$. If $(A,g,a_{0})$ follows all the Peano's axioms, than we can find an isomorphism f between $...
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Are the Inductive Axiom and the Well ordering Principle really equivalent?
I am new to formal math, so apologies if this is naive.
In class, we stated 4 of Peano's axioms. For the fifth, my professor claimed that we may either write the Well Ordering Principle or the ...
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Proof of Edmund Landau's Foundation of Analysis [duplicate]
The theorem is as following:
The proof was split into two parts, namely: Uniqueness and Existence, I have hard time understanding the existence part:
Namely: in the 10-12th line, what did Edmund ...
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3
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Prove if $x \neq 1$ then there exists exactly one $u$ such that $x=u'$
While I'm reading E. Landau's Grundlagen der Analysis (tr. Foundations of Analysis, 1966), I couldn't understand the proof of Theorem 3 at the segment of Natural Numbers which I've quoted below.
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Is there an error in this proof the the "strong induction" theorem? Is this Escherian logic?
By set predicates, I mean the "tests for inclusion" used in defining sets. That may be more of a computer scientific than mathematical use of the the term predicate. I included predicate logic in ...
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In the extension to integers, how should this restriction on the domain of natural numbers be interpreted?
My interpretation of the sub-paragraph in the book which I am asking about is this:
BEGIN SUMMARY
Note that in this context 0 is not considered to be a natural number.
The pairs of natural numbers $...
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4
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249
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Proving $a\ne{b}\implies{a<b}\lor{b<a}$ for natural numbers beginning with 1 using Peano's Axioms without induction hypothesis
Here, I am asking specifically about a proof which does not use an induction hypothesis, and which relies exclusively on Peano's axioms as stated herein. My interest is not in simply producing the ...
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How should this proof of the associativity of natural number addition be understood?
The context of this question is the proofs of the basic laws of arithmetic
beginning with Peano's axioms which are stated starting with the number 1, rather than 0. The modified principle of induction ...
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What is Complete Induction without Hypothesis?: Ordering of $\mathbb{N}$ from Peano's Axioms
The following is from Fundamentals of Mathematics, Volume 1 Foundations of Mathematics: The Real Number System and Algebra; Edited by H. Behnke, F. Bachmann, K. Fladt, W. Suess and H. Kunle
I've used ...
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Proving well ordering principle from PMI (PCI) from peano axioms
Axiomatically, set $\mathbb{N}$ is constructed via injective function $s:\mathbb{N}\rightarrow \mathbb{N}$ and an element $1\in\mathbb{N}$ and we have that $\forall n\in\mathbb{N}:s(n)\neq1$. Together ...
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