Question

Proving $a\ne{b}\implies{a<b}\lor{b<a}$ for natural numbers beginning with 1 using Peano's Axioms without induction hypothesis

score 249 · Answer 1 · 2026-04-15 02:50:46

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

Published on 15 Apr 2026 - 2:50

score 1.2k · Answer 2 · 2026-04-12 15:45:37

1.2k

Views

Peano axioms and commutativity of addition

Published on 12 Apr 2026 - 15:45

score 193 · Answer 3 · 2026-04-11 23:24:46

193

Views

First Order Logic Peano arithmetic Proof

Published on 11 Apr 2026 - 23:24

score 100 · Answer 4 · 2026-05-11 04:10:27

100

Views

What is the pitfall in the inductive "proof" of P(x):= x does not succeed 1?

Published on 11 May 2026 - 4:10

score 648 · Answer 5 · 2026-05-10 23:15:47

648

Views

Consistent theories that proves their own consistence

Published on 10 May 2026 - 23:15

score 643 · Answer 6 · 2026-04-14 18:58:51

643

Views

In Peano's Axioms are the uniqueness of the successor and $x^{\prime}=y^{\prime}\implies{x=y}$ redundant?

Published on 14 Apr 2026 - 18:58

score 131 · Answer 7 · 2026-04-09 06:06:37

131

Views

Decidability of Gödel sentences.

Published on 09 Apr 2026 - 6:06

score 62 · Answer 8 · 2026-04-10 08:16:07

62

Views

Is there a formal term for the algebraic structure equivalent to Thurston's hemigroup without identity?

Published on 10 Apr 2026 - 8:16

score 628 · Answer 9 · 2026-05-11 02:04:31

628

Views

Can we prove the Peano axioms from a type theoretic construction of the natural numbers?

Published on 11 May 2026 - 2:04

score 131 · Answer 10 · 2026-05-10 12:34:52

131

Views

First-order Peano arithmetic and (the lack of) implicit definition of addition

Published on 10 May 2026 - 12:34

score 51 · Answer 11 · 2026-04-15 02:44:34

51

Views

Is this a valid proof that $\forall_{x,y}\left[x\ne y\implies x<y\lor x>y\right]$ in $\mathbb{N}_1$?

Published on 15 Apr 2026 - 2:44

score 324 · Answer 12 · 2026-04-14 17:57:05

324

Views

How would I show that PA proves the godel sentence for PA implies Con(PA)

Published on 14 Apr 2026 - 17:57

score 84 · Answer 13 · 2026-04-10 22:24:03

84

Views

How is it shown using the additive cancellation law that $x\mapsto x+h$ has an inverse?

Published on 10 Apr 2026 - 22:24

score 265 · Answer 14 · 2026-04-12 20:58:15

265

Views

In the extension to integers, how should this restriction on the domain of natural numbers be interpreted?

Published on 12 Apr 2026 - 20:58

score 170 · Answer 15 · 2026-05-10 15:00:37

170

Views

Is it consistent for $P(\mathbb{N})$ to be present in a low layer of the Constructible Universe?

Published on 10 May 2026 - 15:00

List Question

Proving $a\ne{b}\implies{a<b}\lor{b<a}$ for natural numbers beginning with 1 using Peano's Axioms without induction hypothesis

Peano axioms and commutativity of addition

First Order Logic Peano arithmetic Proof

What is the pitfall in the inductive "proof" of P(x):= x does not succeed 1?

Consistent theories that proves their own consistence

In Peano's Axioms are the uniqueness of the successor and $x^{\prime}=y^{\prime}\implies{x=y}$ redundant?

Decidability of Gödel sentences.

Is there a formal term for the algebraic structure equivalent to Thurston's hemigroup without identity?

Can we prove the Peano axioms from a type theoretic construction of the natural numbers?

First-order Peano arithmetic and (the lack of) implicit definition of addition

Is this a valid proof that $\forall_{x,y}\left[x\ne y\implies x<y\lor x>y\right]$ in $\mathbb{N}_1$?

How would I show that PA proves the godel sentence for PA implies Con(PA)

How is it shown using the additive cancellation law that $x\mapsto x+h$ has an inverse?

In the extension to integers, how should this restriction on the domain of natural numbers be interpreted?

Is it consistent for $P(\mathbb{N})$ to be present in a low layer of the Constructible Universe?

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