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

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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)))

^^^
Works for me!

Are there any improvements I could make though?

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.

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:


What is the meaning of ɸ in this context?


What is the meaning of ℕ in this context?


What is the meaning of S in this context?

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

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.

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.

Yes, that is what I intended.

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.

My mathematics abilities are embarrassingly poor, however I have often wondered about this sort of thing. Beyond mathematics, I wonder if all of physical reality could be derived from metaphysical axioms, and indeed, if there are some set of undeniable metaphysical axioms which ontologically necessitate reality as we know (thus answering the question, why is there something instead of nothing).

Yes. Betrand Russell tried to "eliminate all of the paradoxes in mathematics" (ie iron out all of the kinks), by breaking down the entire edifice of western mathematical thought, and then trying to build it all back up again on a new foundation. And tried to do that in a book that talks about mathematics for 169 pages before it gets to the concept of "one and one equals two"!

He didn't succeed. And later Godel proved it impossible.

So I guess that even math cant be based totally upon logic.

I am probably one of the few people you will come across gabemai314 that can actually give you sound advice on this topic. Your system is one of an infinite number of other possible systems that Mathematicians have discovered and explored.

All your symbols were intelligible to me except the phi. How did you type your equations? ASCII codes? Impressed with the work it must have taken. Fun to see the naturals, real and rational sets defined using such elaborate set based notation. Since your real and rational sets are defined relative to the natural set they should not be included in the "axioms" of your system.

OK, now for the harsh words unfortunately. Before that though, if you want to go over the details of your derivations as a side topic, I would be more than willing to discuss that with you. I am curious about this part:

Not sure if this part works. The rest of your axioms seem fine, even comprehensive, but real numbers are not as easy to define by defining them relative to other sets. It looks like you are trying to use the definition of a continuous limit with the well-known ε - δ definition, but I am not sure if the equation above is even syntactically correct, which is bothering.

I think you should look up Dedekind Cuts.

I don't think your system could possibly encapsulate all of Math, but I like the initiative. The modern day approach to this is Homotopy Type Theory.

For myself, the whole of math should be derivable from the binary set {true, false} alone if you are trying to really pair things down.

Good luck either way!

It cannot be done and this is well known in computer science and logic.

https://plato.stanford.edu/entries/goed ... pleteness/
(it's a long read in its entirety)

Cool, someone who knows the math biz Nice. I would be all, well, duuhh... but then, I did not put in a winky emoticon at the end of the line you quoted; so let the record note from here on out that I said it as a joke. I think that @naturalplastic earlier referred to this kind of critique as well... HoTT takes the incompleteness critique much futher though. This is why I wrote:



There are supposedly uncountable numbers of self-consistent Set Theories, ZFC systems and things much more exotic still (e.g. self-consistent sets of groupoid axioms, generalizations of p-adic number system axioms, etc...).

Even so, I think everyone would have to admit that a lot of Mathematics centers around decibility: is it true, or is it false? Plus, do you know how far you can take 0 and 1? Pretty damn far is the answer.

Although, of late, I have been thinking about Computational Proof Theory and how it relates to Mathematics, as a whole. I really wonder in some ways whether the diagonal slash method of proof and some related proof ideas might not actually turn out to be wrong in a way that Douglas Richard Hofstadter in his book "Gödel, Escher, Bach: An Eternal Golden Braid", would have referred to as 'mu'.

Let's define the word "mu" to mean the mathematical equivalent of "not even wrong". This word comes from Japanese and Korean and means 'not, without'. Wu is the Chinese equivalent, but sounds like woo, which is taken. It is Hofstadter's Zen suggestion for logical take back, "unask the question".

Like having a question that confuses mutually exclusive categories of things. example: "Tommorow is Tuesday, that means it can be the month of May, right?" Well, yes and no... confusingly so! Sorry, best example I could think up.

Consider Russell's Paradox (which ZFC and type systems were created to address). Let ℕ* be defined as follows, where Pow means find the power set, power set being defined as the set of all subsets of a set.

ℕ* = { n : n ∈ ℕ U Pow(ℕ*) }

The following is a set that should have all of the same problems as that proposed by the original set of Russell.

Let

R = { x : x ∈ ℕ* ∧ x ∉ x },

then for a given set, a "program" can be written to determine if an element is in the set R, or not. This would seem to invalidate Russell's Paradox.

If you accept the self-refferential definition of ℕ*, which, let's be honest, you could quibble about, then the next thing that might be going through your head is, is this guy actually saying Russell's Paradox is invalid? Maybe. I am still considering it myself.

The problem is in the type information and how sets are looked at. Let's do this in two steps.

Step 1. Reframe your thinking about sets. Let a set not be a collection of things, but instead be a function that takes in an element from a well-defined universal set as input, returning true if the element is a member of the set, or false if is not.

Let

ℤ₂ = { 0, 1 }

for future reference, 0 false, 1 true.

Step 2. Some minimal Type theory notation needs explaining. Another way to rethink set theory is by using types. You first define a finite alphabet of symbols. Ordered, linear arrangements of these alphabet symbols are called strings. If a string meets the condition of having a certain format then that makes it a 'term' of that 'type' of string. Using Capitol Roman letters for Type, a function that takes a type T term and produces a type P term is given by

T -> P. (not same arrow as implies below, reads as T maps to P)

Let's say we want to denote a type that can be either P or Q. This is given by

P + Q

A set that has no subsets and whose elements are of type T is given by the type

T -> ℤ₂

Now for the finale, assuming you have understood the previous codswallop. The original problems with Russell's paradox are two-fold.

1. Mu on the question R ∈ R or R ∉ R. The wrong type is being used. Mu on this question in general, because except when used in the body of a set definition, this statement has a false hidden assumption.

Looking at the sets in question from the function / type theoretic point of view, the set R is of type

R = ℕ* -> ℤ₂

The type of the set that tests for subsets of all orders is given by

P = (P + R + (R -> ℤ₂) + ((R -> ℤ₂) -> ℤ₂) + ...) -> ℤ₂,

and as

R ≠ P, as Johnnie Cochran would have said: if the glove does not fit, then you must aquit. The type given by P is the Ouroboros of the problem. No matter how hard you try to finitely redefine R to include more types, P comes along and swallows it's own tail, making your efforts pointless.

2. For the set R as previously defined, it is the case that

¬(x ∉ R → x ∈ R) ∧ ¬(x ∈ R → x ∉ R),

thus obviating the paradox completely. The reason for the above is what I would take as the second problem in Russell's Paradox. I can already feel the mathematicians who read this might be fuming at the above. They know that if you just look in the body of the definition of R that I am lying, but am I?

The issue now is, when does one stop computing?

Say you determine an element is in a set, do you stop there and leave it at that, or do you then "check" again to make sure after that the computation that everything is self-consistent? The idea of us checking to make sure everything is all right, could in itself, be a misunderstanding.

Consider what happens in the usual Cantor proof type of argument. Let

x ∈ R,

then x ∉ R, but then x ∈ R, and so on, and so forth. It's a never ending paradox, Bastian turns into Atreyu riding Falkor who eats his own tail and the rest of Fantasia along with him. No more allusions from here on out, I promise.

But if you look at it as a computer program, when x ∈ R = 1, that is the end of the computation. You don't compute again to check anything, the computation is done and nothing further should be inferred from it.

Note on infinity. It drives you kind of batty thinking too deeply about it, so be cautious. Cantor died in a mental assylum sanatorium. I chose ℕ as a set to base R on because it is countable. You could complain, what about the infinitely infinite set of other higher transfinite ordinals? Yes, that is also an avenue of discovery / criticism, but going down the road of trying to constrain infinity makes you chase your own tail, driving you Craaazy! I made headway at all, so I will leave it at that. I don't want to die in an assylum.

Infinity just can not be conceptually encompassed, no matter how hard we try. Don't worry, Cthulhu loves you, so all is fine, I assure you

This Post is Dedicated to the memory of Cantor. Look him up if you are ignorant of this man :0

Cool, someone who knows the math biz Nice. I would be all, well, duuhh... but then, I did not put in a winky emoticon at the end of the line you quoted; so let the record note from here on out that I said it as a joke. I think that @naturalplastic earlier referred to this kind of critique as well... HoTT takes the incompleteness critique much futher though. This is why I wrote:



There are supposedly uncountable numbers of self-consistent Set Theories, ZFC systems and things much more exotic still (e.g. self-consistent sets of groupoid axioms, generalizations of p-adic number system axioms, etc...).

Even so, I think everyone would have to admit that a lot of Mathematics centers around decibility: is it true, or is it false? Plus, do you know how far you can take 0 and 1? Pretty damn far is the answer.

Although, of late, I have been thinking about Computational Proof Theory and how it relates to Mathematics, as a whole. I really wonder in some ways whether the diagonal slash method of proof and some related proof ideas might not actually turn out to be wrong in a way that Douglas Richard Hofstadter in his book "Gödel, Escher, Bach: An Eternal Golden Braid", would have referred to as 'mu'.

Let's define the word "mu" to mean the mathematical equivalent of "not even wrong". This word comes from Japanese and Korean and means 'not, without'. Wu is the Chinese equivalent, but sounds like woo, which is taken. It is Hofstadter's Zen suggestion for logical take back, "unask the question".

Like having a question that confuses mutually exclusive categories of things. example: "Tommorow is Tuesday, that means it can be the month of May, right?" Well, yes and no... confusingly so! Sorry, best example I could think up.

Consider Russell's Paradox (which ZFC and type systems were created to address). Let ℕ* be defined as follows, where Pow means find the power set, power set being defined as the set of all subsets of a set.

ℕ* = { n : n ∈ ℕ U Pow(ℕ*) }

The following is a set that should have all of the same problems as that proposed by the original set of Russell.

Let

R = { x : x ∈ ℕ* ∧ x ∉ x },

then for a given set, a "program" can be written to determine if an element is in the set R, or not. This would seem to invalidate Russell's Paradox.

If you accept the self-refferential definition of ℕ*, which, let's be honest, you could quibble about, then the next thing that might be going through your head is, is this guy actually saying Russell's Paradox is invalid? Maybe. I am still considering it myself.

The problem is in the type information and how sets are looked at. Let's do this in two steps.

Step 1. Reframe your thinking about sets. Let a set not be a collection of things, but instead be a function that takes in an element from a well-defined universal set as input, returning true if the element is a member of the set, or false if is not.

Let

ℤ₂ = { 0, 1 }

for future reference, 0 false, 1 true.

Step 2. Some minimal Type theory notation needs explaining. Another way to rethink set theory is by using types. You first define a finite alphabet of symbols. Ordered, linear arrangements of these alphabet symbols are called strings. If a string meets the condition of having a certain format then that makes it a 'term' of that 'type' of string. Using Capitol Roman letters for Type, a function that takes a type T term and produces a type P term is given by

T -> P. (not same arrow as implies below, reads as T maps to P)

Let's say we want to denote a type that can be either P or Q. This is given by

P + Q

A set that has no subsets and whose elements are of type T is given by the type

T -> ℤ₂

Now for the finale, assuming you have understood the previous codswallop. The original problems with Russell's paradox are two-fold.

1. Mu on the question R ∈ R or R ∉ R. The wrong type is being used. Mu on this question in general, because except when used in the body of a set definition, this statement has a false hidden assumption.

Looking at the sets in question from the function / type theoretic point of view, the set R is of type

R = ℕ* -> ℤ₂

The type of the set that tests for subsets of all orders is given by

P = (P + R + (R -> ℤ₂) + ((R -> ℤ₂) -> ℤ₂) + ...) -> ℤ₂,

and as

R ≠ P, as Johnnie Cochran would have said: if the glove does not fit, then you must aquit. The type given by P is the Ouroboros of the problem. No matter how hard you try to finitely redefine R to include more types, P comes along and swallows it's own tail, making your efforts pointless.

2. For the set R as previously defined, it is the case that

¬(x ∉ R → x ∈ R) ∧ ¬(x ∈ R → x ∉ R),

thus obviating the paradox completely. The reason for the above is what I would take as the second problem in Russell's Paradox. I can already feel the mathematicians who read this might be fuming at the above. They know that if you just look in the body of the definition of R that I am lying, but am I?

The issue now is, when does one stop computing?

Say you determine an element is in a set, do you stop there and leave it at that, or do you then "check" again to make sure after that the computation that everything is self-consistent? The idea of us checking to make sure everything is all right, could in itself, be a misunderstanding.

Consider what happens in the usual Cantor proof type of argument. Let

x ∈ R,

then x ∉ R, but then x ∈ R, and so on, and so forth. It's a never ending paradox, Bastian turns into Atreyu riding Falkor who eats his own tail and the rest of Fantasia along with him. No more allusions from here on out, I promise.

But if you look at it as a computer program, when x ∈ R = 1, that is the end of the computation. You don't compute again to check anything, the computation is done and nothing further should be inferred from it.

Note on infinity. It drives you kind of batty thinking too deeply about it, so be cautious. Cantor died in a mental assylum sanatorium. I chose ℕ as a set to base R on because it is countable. You could complain, what about the infinitely infinite set of other higher transfinite ordinals? Yes, that is also an avenue of discovery / criticism, but going down the road of trying to constrain infinity makes you chase your own tail, driving you Craaazy! I made headway at all, so I will leave it at that. I don't want to die in an assylum.

Infinity just can not be conceptually encompassed, no matter how hard we try. Don't worry, Cthulhu loves you, so all is fine, I assure you

This Post is Dedicated to the memory of Cantor. Look him up if you are ignorant of this man :0

Sorry for posting twice, not my intent... what is up with the captcha yall?