All Questions
Tagged with well-orders peano-axioms
7
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Confusion about the validity of the proof of Trichotomy of order for natural numbers in Tao's Analysis
It's well-known that in Tao's Analysis I P28, he provides a provement of Trichotomy of order for natural numbers as follows.
Denote the number of correct propositions among the three (i.e. $a<b,\ ...
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80
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Does every well-ordered set obeying the non-induction Peano axioms have a well-ordering compatible with the successor operation?
Let $N$ be a well-ordered set together with a unary operation $s$ that obeys the following axioms (they are just the Peano axioms without induction):
$0 \in N$
for each $n \in N$ we have $s(n) \in N$
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What is the definition of the "order" of well order property in Peano Axioms?
For Peano Axiom, mathematical induction is equivalent to well order property.
But in well order property, what is the definition of "order"?
In detail, if we define the "order" $b\...
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100
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Prove strict well-order from Peano successor function
Is there a way to prove from (a variation) of the Peano successor function without induction that < is a strict well-order? Specifically, let $m\leftarrowtail n$ represent that $n$ is the successor ...
- 555
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Proving Natural Predecessor existence using Well Order
I know that the Principle of Induction is equivalent to Well Order at least in Natural Numbers, but I have seen that the demonstration of Induction using Well Order uses the existence of Predecessor:
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Defining the Order of the Natural Numbers
Is there a way to give the regular partial order of the natural number directly from their definition through the Infinity Axiom? I have only ever seen the partial order of the natural number to be ...
- 97
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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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