All Questions
Tagged with peano-axioms elementary-set-theory
42
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not precisely understanding what is asked by ex 3.5.13 Tao Analysis I (only one natural number system)
I am failing to understand the intent of the question posed by exercise 3.5.13 of Tao's Analysis I 4th ed.
The purpose of this exercise is to show that there is essentially only one version of the ...
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Rigorous proof that cardinality of a disjoint union is the sum of cardinalities for finite sets
In a lot of books there are intuitive(but sort of hand wavy) proofs for finite, disjoint sets $A$ and $B$ that
$$
|A \cup B| = |A| + |B|
$$
since $A = \{a_1,a_2,a_3,\cdots,a_{|A|}\}$ and $B = \{b_1,...
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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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Applying Peano Axioms to Subsets of Natural Numbers [duplicate]
Concise Question
Is it possible to apply the Peano axioms to subsets of the natural numbers and thus define arithmetic operations on those subsets strictly in terms of those subsets? If so, is it also ...
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How to prove that there are $n$ natural numbers that are less or equal than $n$ and what properties are allowed to use in induction.
Let $n \in \mathbf{N}$. I wondered how to prove that there are exactly $n$ natural numbers that are smaller or equal than $n$. This seems somewhat circular which confuses me. I guess the way to do ...
- 389
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Finiteness, finite sets and representing its elements.
A set $S$ is called finite if there exists a bijection from $S$ to $\{1,...,n\}$ for one $n \in \mathbf{N}$. It is then common to write its elements as $s_1,...,s_n$. I now wonder, why this is ...
- 389
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3
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Can the natural numbers contain an element that is not representable by a number?
I read the following document: https://www.math.wustl.edu/~freiwald/310peanof.pdf . In this document, the author wants to formalize that natural numbers, that are informally thought of as a collection ...
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Peano Axioms and modeling sets.
I am assuming the standard Peano-Axioms, which can be found here https://en.wikipedia.org/wiki/Peano_axioms under "Formulation". Does one assume that a set of objects called $\mathbf{N}$ ...
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How to construct a partial ordering from Peano's 5 Axioms?
I am trying to formally construct the usual partial ordering LTE from Peano's 5 Axioms. Would the following construction work?
$$\forall a,b: [(a,b)\in LTE \iff(a,b)\in N^2$$
$$\land ~ \forall c\...
- 14.8k
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$\pi$ is isomorphism from one Peano system $(N, S, e)$ to another $(N', S', e')$, then $\pi^{-1}$ is isomorphism from $(N', S', e')$ to $(N, S, e)$
This is an exercise from Cunningham's book "Set Theory: A First Course".
Theorem: Let $(N, S, e)$ and $(N', S', e')$ be Peano systems. Let $\pi$ be an isomorphism from $(N, S, e)$ onto $(N', ...
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Proof critique of least number principle, please!
I am independently working through Elliot Mendelson's "Number Systems and the Foundations of Analysis," which I find very well-written and rigorous, along with providing a healthy set of ...
- 368
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418
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Prove: if $m \in n$ then $m^+ \subseteq n$.
The following is exercise 5(d), section 6.2, from A book of set theory, by Charles Pinter (pg. 122).
5. Prove the following, where $m, n, p \in \omega$.
d) If $m \in n$, then $m^+ \subseteq n$.
...
user837396
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Show that $(P,S,1)$ is a Peano system if and only if it satisfies the Iteration Theorem
You can skip the first section(hints and useful things) if you want, I will keep editing this along I got more findings related to the question.
Hints and useful things: I find out that the book from ...
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How to show that a triple $(P, S, 1)$ constitutes a Peano System?
Mendelson (in Number Systems & the Foundations of Analysis) defines a Peano System as a triple $(P, S, 1)$ consisting of a set $P$, a distinguished element $1 \in P$, and a singulary operation $S :...
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Determine whether or not the following structure $(P,S,1)$ is a Peano System
First this is how the book define as a Peano System.
By a Peano System we mean a set $P$, a particular element $1$ in $P$, and a singulary operation $S$ on $P$ such that the following axioms are ...