# Neo-logicism, set theory, and first-order Peano arithmetic

Page 1 of 1 [ 9 posts ]

gabemai314
Blue Jay

Joined: 12 Feb 2017
Age: 20
Gender: Male
Posts: 75
Location: Loveland, Colorado, USA, Earth, Solar System, Gould Belt, Orion Arm, Milky Way Galaxy, Local Group, Virgo Supercluster, Laniakea Supercluster, Universe

31 May 2019, 5:51 am

Hello all! I was wondering what people think of the neo-logicism, the idea that all of mathematics can be derived from logic. I have outlined a potential formulation of this idea below. Please critique and offer suggestions.

((x∧y)∨(¬x∧¬y))→x=y
((x∨y)∧(¬x∨¬y))→x≠y
U={x|ɸ(x)}
∀x∈U(φ(x))≡(U→φ(x))
∃x∈U(φ(x))≡({x∈U|φ(x)}≠∅}
∀x∈ℕ(x=x)
∀x,y∈ℕ(x=y→y=x)
∀x,y,z∈ℕ((x=y∧y=z)→x=z)

∀x∈ℕ(0≠S(x))
∀x,y∈ℕ(S(x)=S(y)→x=y)
∀x∈ℕ(x+0=x)
∀x,y∈ℕ(x+S(y)=S(x+y))
∀x,y∈ℕ(x-y≡∃z∈ℕ(x=y+z))
∀x∈ℕ(x*0=0)
∀x,y∈ℕ(x*S(y)=x*y+x)
∀x,y∈ℕ(x/y≡∃z∈ℕ(x=y*z))
∀x,y∈ℕ(x≤y→∃z∈ℕ(x+z=y))
∀x,y∈ℕ(x<y≡(x≤y∧x≠y))
∀x,y,z∈ℕ((x+y)+z=x+(y+z))
∀x,y∈ℕ(x+y=y+x)
∀x,y,z∈ℕ((x*y)*z=x*(y*z))
∀x,y∈ℕ(x*y=y*x)
∀x,y,z∈ℕ(x*(y+z)=(x*y)+(x*z))
∀x,y,z∈ℕ((x<y∧y<z)→x<z)
∀x∈ℕ(¬(x<x))
∀x,y∈ℕ(x<y∨x=y∨y<x)
∀x,y,z∈ℕ(x<y→x+z<y+z)
∀x,y,z∈ℕ((0<z∧x<y)→x*z<y*z)
0<1∧∀x∈ℕ(x>0→x≥1)
∀x∈ℕ(x≥0)

ℤ={(a,b)∈ℕ|b≤a→(a-b)∨a<b→-(b-a)}
∀(a,b),(c,d)∈ℕ((a,b)~(c,d)≡a+d=b+c)
[(a,b)]+[(c,d)]=[(a+c,b+d)]
[(a,b)]-[(c,d)]=[(a+d,b+d)]
[(a,b)]<[(c,d)]≡a+d<b+c

ℚ={(m,n)∈ℤ|n≠0}
∀(m,n)∈ℤ((m₁,n₁)~(m₂,n₂)≡m₁n₂=m₂n₁)
(m₁,n₁)+(m₂,n₂)=(m₁n₂+m₂n₁,n₁n₂)
(m₁,n₁)-(m₂,n₂)=(m₁n₂-m₂n₁,n₁n₂)
(m₁,n₁)*(m₂,n₂)=(m₁m₂,n₁n₂)
(m₁,n₁)/(m₂,n₂)=(m₁n₂,m₂n₁)
(m₁,n₁)≤(m₂,n₂)≡((n₁n₂>0→m₁n₂≤m₂n₁)∨(n₁n₂<0→m₁n₂≥m₂n₁))

ℝ={xᵢ∈ℚ|∀ε∈ℚ(ε>0),∃N∈ℤ(∀m,n∈ℕ((m∧n)>N∧|xₘ-xₙ|<ε))}
(xₙ)+(yₙ)=(xₙ+yₙ)
(xₙ)*(yₙ)=(xₙ*yₙ)
(xₙ≤yₙ)≡((x~y)∨∃N∈ℤ(∀n>N(xₙ≤yₙ)))

_________________
I am open to PMs.
GitHub: https://github.com/gabemaiberger/

naturalplastic
Veteran

Joined: 26 Aug 2010
Age: 64
Gender: Male
Posts: 20,669
Location: temperate zone

31 May 2019, 5:53 pm

^^^
Works for me!

gabemai314
Blue Jay

Joined: 12 Feb 2017
Age: 20
Gender: Male
Posts: 75
Location: Loveland, Colorado, USA, Earth, Solar System, Gould Belt, Orion Arm, Milky Way Galaxy, Local Group, Virgo Supercluster, Laniakea Supercluster, Universe

01 Jun 2019, 2:02 am

naturalplastic wrote:
^^^
Works for me!

Are there any improvements I could make though?

_________________
I am open to PMs.
GitHub: https://github.com/gabemaiberger/

naturalplastic
Veteran

Joined: 26 Aug 2010
Age: 64
Gender: Male
Posts: 20,669
Location: temperate zone

02 Jun 2019, 2:25 am

I was just making a joke about how incomprehensible your topic is to most folks, including me.

To me you might as well be translating Swahili into Eskimo language.

There are many math experts on this site, but even they haven't responded to you yet.

Mona Pereth
Veteran

Joined: 11 Sep 2018
Age: 61
Gender: Female
Posts: 915
Location: New York City (Queens)

09 Jun 2019, 7:15 pm

gabemai314 wrote:
Hello all! I was wondering what people think of the neo-logicism, the idea that all of mathematics can be derived from logic. I have outlined a potential formulation of this idea below. Please critique and offer suggestions.

I took several courses in logic and set theory years ago. I'm a bit rusty on this stuff, but I might be able to give you some useful feedback if you answer a few questions about your notation:

gabemai314 wrote:
((x∧y)∨(¬x∧¬y))→x=y
((x∨y)∧(¬x∨¬y))→x≠y
U={x|ɸ(x)}

What is the meaning of ɸ in this context?

gabemai314 wrote:
∀x∈U(φ(x))≡(U→φ(x))
∃x∈U(φ(x))≡({x∈U|φ(x)}≠∅}
∀x∈ℕ(x=x)

What is the meaning of ℕ in this context?

gabemai314 wrote:
∀x,y∈ℕ(x=y→y=x)
∀x,y,z∈ℕ((x=y∧y=z)→x=z)
∀x∈ℕ(0≠S(x))

What is the meaning of S in this context?

_________________
- Finally diagnosed with ASD in May 2019, after having suspected it for over ten years, and after having deeply explored the autism community for over one year while waiting for and obtaining diagnosis.
- In longterm relationship with boyfriend who was diagnosed with Asperger's Syndrome in 2001.
- Long history of participation in various oddball subcultures.
- My "Getting to know each other" thread: Hello from NYC.

Mona Pereth
Veteran

Joined: 11 Sep 2018
Age: 61
Gender: Female
Posts: 915
Location: New York City (Queens)

10 Jun 2019, 3:32 am

Also I'm a bit confused by your syntax here:

gabemai314 wrote:
∀x∈U(φ(x))≡(U→φ(x))
∃x∈U(φ(x))≡({x∈U|φ(x)}≠∅}
∀x∈ℕ(x=x)

Is the above intended to mean the same things as the following?

((∀x∈U)(φ(x)) ≡ (U→φ(x))
((∃x∈U)(φ(x))) ≡ ({x∈U|φ(x)}≠∅})
(∀x∈ℕ)(x=x)

_________________
- Finally diagnosed with ASD in May 2019, after having suspected it for over ten years, and after having deeply explored the autism community for over one year while waiting for and obtaining diagnosis.
- In longterm relationship with boyfriend who was diagnosed with Asperger's Syndrome in 2001.
- Long history of participation in various oddball subcultures.
- My "Getting to know each other" thread: Hello from NYC.

gabemai314
Blue Jay

Joined: 12 Feb 2017
Age: 20
Gender: Male
Posts: 75
Location: Loveland, Colorado, USA, Earth, Solar System, Gould Belt, Orion Arm, Milky Way Galaxy, Local Group, Virgo Supercluster, Laniakea Supercluster, Universe

10 Jun 2019, 8:13 pm

Mona Pereth wrote:
gabemai314 wrote:
Hello all! I was wondering what people think of the neo-logicism, the idea that all of mathematics can be derived from logic. I have outlined a potential formulation of this idea below. Please critique and offer suggestions.

I took several courses in logic and set theory years ago. I'm a bit rusty on this stuff, but I might be able to give you some useful feedback if you answer a few questions about your notation:

gabemai314 wrote:
((x∧y)∨(¬x∧¬y))→x=y
((x∨y)∧(¬x∨¬y))→x≠y
U={x|ɸ(x)}

What is the meaning of ɸ in this context?

gabemai314 wrote:
∀x∈U(φ(x))≡(U→φ(x))
∃x∈U(φ(x))≡({x∈U|φ(x)}≠∅}
∀x∈ℕ(x=x)

What is the meaning of ℕ in this context?

gabemai314 wrote:
∀x,y∈ℕ(x=y→y=x)
∀x,y,z∈ℕ((x=y∧y=z)→x=z)
∀x∈ℕ(0≠S(x))

What is the meaning of S in this context?

ɸ(x) is the function bounding the domain of discourse, ℕ is the set of natural numbers, and S(x) is the successor function for the natural numbers where S(x)=x+1.

_________________
I am open to PMs.
GitHub: https://github.com/gabemaiberger/

gabemai314
Blue Jay

Joined: 12 Feb 2017
Age: 20
Gender: Male
Posts: 75
Location: Loveland, Colorado, USA, Earth, Solar System, Gould Belt, Orion Arm, Milky Way Galaxy, Local Group, Virgo Supercluster, Laniakea Supercluster, Universe

10 Jun 2019, 8:16 pm

Mona Pereth wrote:
Also I'm a bit confused by your syntax here:

gabemai314 wrote:
∀x∈U(φ(x))≡(U→φ(x))
∃x∈U(φ(x))≡({x∈U|φ(x)}≠∅}
∀x∈ℕ(x=x)

Is the above intended to mean the same things as the following?

((∀x∈U)(φ(x)) ≡ (U→φ(x))
((∃x∈U)(φ(x))) ≡ ({x∈U|φ(x)}≠∅})
(∀x∈ℕ)(x=x)

Yes, that is what I intended.

_________________
I am open to PMs.
GitHub: https://github.com/gabemaiberger/

Tufted Titmouse

Joined: 15 Jun 2017
Gender: Male
Posts: 41
Location: Latin America

12 Jul 2019, 9:07 pm

The original logicism project was proposed by Bertrand Russell more than a century ago. After some years, the idea to derive all the mathematics from logic was finally proved impossible by Kurt Gödel and his two Incompleteness Theorems.

In consequence, there aren't and can't be exact foundations for mathematics. Yet, the most accepted axioms used to give 'foundations' of mathematics are Peano's axiomatization of natural numbers (mostly used in analysis) and, for other way, the axiomatization of Zermelo-Fraenkel for sets (mostly used in areas like logic and algebra).

You can search for both of them in the Inernet or in academic books.

I hope I have helped, sorry for -maybe- bad english.

_________________
¡Look me to the eyes!