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

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gabemai314
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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)]=[(ac+bd,ad+bc)]
[(a,b)]/[(c,d)]=[(ac-bd,ad-bc)]
[(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)))


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naturalplastic
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31 May 2019, 5:53 pm

^^^
Works for me! :D



gabemai314
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01 Jun 2019, 2:02 am

naturalplastic wrote:
^^^
Works for me! :D

Are there any improvements I could make though?


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naturalplastic
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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. :lol:

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



Mona Pereth
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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?


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Mona Pereth
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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)

If not, please clarify your original versions of these by adding more parentheses where appropriate.


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- In longterm relationship with boyfriend who was diagnosed with Asperger's Syndrome in 2001.
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gabemai314
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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.


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gabemai314
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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)

If not, please clarify your original versions of these by adding more parentheses where appropriate.


Yes, that is what I intended.


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samuraivader
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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.


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