Questions tagged [peano-axioms]
For questions on Peano axioms, a set of axioms for the natural numbers.
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On the consistency of satisfiable first order theories
Considering this question, we know that a first order theory that admits a model has to be consistent.
A model for a theory $T$ in a language $\mathcal L$ is an interpretation of $\mathcal L$ in which ...
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Why define addition with successor?
I'm reading Russell's Introduction to Mathematical Philosophy Russell defines the sum of two numbers in terms of successors. I don't understand why:
Suppose we wish to define the sum of two numbers. ...
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Peano’s Fifth axiom as stated by Russell
I’m translating Russell’s Introduction to Mathematical Philosophy. I’m having difficulty understanding his formulation of the fifth axiom:
(5) Any property which belongs to 0, and also to the ...
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On the uniqueness of the addition operation on $\mathbb{N}$
My textbook (Amann and Escher, Analysis I) gives a theorem which says that the operations of addition and multiplication (and a partial order $\leq$) exist and are uniquely defined by a whole host of ...
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Why universal closure?
In logic, why do we talk of universal closure of a formula, and don't consider its "existential closure" (as far as I know)?
I guess that one of the reasons may be that interesting systems ...
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Is my understanding of Peano Axioms as mentioned below correct? I’ll also be grateful if questions below are answered definitively [closed]
I have concluded the reading of second chapter of Prof. Tao’s Analysis books in which he covers natural numbers and defines addition and multiplication operation on them,
He states the following ...
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What is 'increment' in Peano Axioms?
I am reading Tao's book on Analysis in which the first two axioms apropos natural numbers are,
0 is a natural number.
If n is a natural number, then n++ is also a natural number.
As a motivation ...
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Irregular Induction Theorem for $\mathbb{N}\times\mathbb{N}$
I am trying to prove this irregular induction theorem that would help prove a recursion theorem I am working on.
Can you help? Here is the theorem:
$\forall X (\forall x \in \mathbb{N} (\langle x,0 \...
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Are the axioms of analysis a combination of Peano axioms and set theory axioms?
Observing the use of mathematical induction in proving the finite version of the axiom of choice, I began to ponder. Why is mathematical induction from Peano axioms employed to prove facts about sets? ...
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How to prove natural number addition using induction? [duplicate]
I am a self learner so excuse me if I am asking a seemingly easy question , But I ve been stuck at this point for couple of days , I think I understand mathematical induction and what the author is ...
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Is there a concept of finiteness independent of the successor function?
Why is there no infinite natural number, and why does finiteness need to be closed under the successor function?
I think can understand why something like $…S(S(0))…$ is not a natural number because ...
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Gödels incompleteness theorem false for natural numbers
Do I understand it correctly that in an axiomatic system that includes the natual numbers defined by using Peano's axiom, where the 9th axiom (induction) is formulated using 2nd-order logic, then
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Does the property $T\vdash Pvbl_T(\ulcorner \sigma \urcorner) \implies T\vdash \sigma$ apply to set theories?
I know from other posts that $PA\vdash Pvbl_{PA}(\ulcorner \sigma \urcorner ) \implies PA\vdash \sigma$ and this applies to other extensions/restrictions of PA as well. Does it also apply to set ...
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Analysis I, can Tao's construction of the integers be further simplified?
In chapter 4 of Analysis I by Terence Tao, we have the following note about the set theoretic construction of the integers:
In the language of set theory, what we are doing here is starting with the ...
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Circular reasoning when explaining positional notation
I am trying to explain from scratch the foundation of mathematics and couldn't start anywhere but from Peano's axioms.
After that I introduced the set of digits = {1, 2, 3, 4, 5, 6, 7, 8, 9} where ...
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