Boolean Logic cannot deal with infinitely many objects

[Hurkyl]

What is a property of a set?

Is it the case that $\forall a : a \in \{\_\}$?

 

[phoenixthoth]

Originally posted by Hurkyl
I think that if you're going to talk about a universal set, you're going to need to explicitly demonstrate why the usual proof of $X \neq \mathcal{P}(X)$ fails, and how the usual constructions of paradoxes fails. (such as the set of all sets that don't contain themselves)

well, that the set of sets that don't contain themselves is russell's paradox. the point is that, as i argued in the thread "russell's paradox: the achilles heal of solopsism," if you use 3-valued logic, then the tautology used to argue by contradiction is no longer a tautology. furthermore, a way around 3-valued logic is to state that in the subsets axiom, if a well formed formula leads to a contradiction, then that is not a subset.

let $$U$$ be the universal set. i will show how cantor's diagonal argument fails to show that $$U$$ does not have a map onto $$P(U)$$. let $$id_{U}$$ be the identity map on $$U$$; i will show that $$id_{U}$$ maps onto $$P\left( U \right)$$ if contradiction is not a tautology (which it isn't in 3-valued logic).

first of all, note that since $$\forall x\left( x\in U\right)$$, it follows that if $$x$$ is a set then $$x\subset U$$, hence $$P\left( U\right) \subset U$$ so it makes sense to think of $$id_{U}$$ as having $$P\left( U\right)$$ involved in the range.

let's follow cantor's diagonal argument, applied to $$id_{U}$$.

$$D_{id_{U}}:=\left\{ x\in U:x\notin id_{U}\left( x\right) \right\}$$. but by what $$id_{U}$$ is, this means that $$D_{id_{u}}=\left\{ x\in U:x\notin x\right\}$$. note that $$D_{id_{U}}\in P\left( U\right)$$. then if $$id_{U}$$ is onto, $$id_{U}\left( x^{\ast }\right) =D_{id_{U}}$$ for some $$x^{\ast }\in U$$. hence, $$x^{\ast }=\left\{ x\in U:x\notin x\right\}$$. now we have a contradiction in two valued logic: $$x^{\ast }\in x^{\ast }\leftrightarrow x^{\ast }\notin x^{\ast }$$.

this argument is based on a tautology called "argument by contradiction" and this tautology is no longer a tautology in 3-valued logic.

one can show that if there is a 1-1 map from $$U$$ to $$x$$, then $$U=x$$. hence, $$U=P\left( U\right)$$. also, if there is a function from $$x$$ onto $$U$$, then $$x=U$$.

 

[Hurkyl]

In what I have read (long ago) on multivalued logics, the classical paradoxes in binary logic can fairly straightforwardly be extended to multi-valued logic.

For instance, $S := \{ x | x \in x\; \mbox{is not true} \}$ sufficies for at least one ternary logic.

if a well formed formula leads to a contradiction, then that is not a subset.

And there's the rub; we need to know what a "safe" set of formulas is. This is a purely metamathematical concern; I can't see any way it could be written formally.


Anyways, it is an interesting exercise to formally write up your ternary logic and see if it really sufficies. I bet that replacing "$p \notin q$" in Cantor's argument with "$p \in q$ is false or <name of third logical value>", then you can still derive a contradiction.

(actually, my gut tells me that you don't even need this modification; but I can't be sure until I see just what your ternary logic looks like)

 

[moshek]

The 6 th' problem of Hilbert ( 1900 Paris)

6. Mathematical treatment of the axioms of physics
The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which mathematics plays an important part; in the first rank are the theory of probabilities and mechanics.

As to the axioms of the theory of probabilities,14 it seems to me desirable that their logical investigation should be accompanied by a rigorous and satisfactory development of the method of mean values in mathematical physics, and in particular in the kinetic theory of gases.

Important investigations by physicists on the foundations of mechanics are at hand; I refer to the writings of Mach,15 Hertz,16 Boltzmann17 and Volkmann. 18 It is therefore very desirable that the discussion of the foundations of mechanics be taken up by mathematicians also. Thus Boltzmann's work on the principles of mechanics suggests the problem of developing mathematically the limiting processes, there merely indicated, which lead from the atomistic view to the laws of motion of continua. Conversely one might try to derive the laws of the motion of rigid bodies by a limiting process from a system of axioms depending upon the idea of continuously varying conditions of a material filling all space continuously, these conditions being defined by parameters. For the question as to the equivalence of different systems of axioms is always of great theoretical interest.

If geometry is to serve as a model for the treatment of physical axioms, we shall try first by a small number of axioms to include as large a class as possible of physical phenomena, and then by adjoining new axioms to arrive gradually at the more special theories. At the same time Lie's a principle of subdivision can perhaps be derived from profound theory of infinite transformation groups. The mathematician will have also to take account not only of those theories coming near to reality, but also, as in geometry, of all logically possible theories. He must be always alert to obtain a complete survey of all conclusions derivable from the system of axioms assumed.

Further, the mathematician has the duty to test exactly in each instance whether the new axioms are compatible with the previous ones. The physicist, as his theories develop, often finds himself forced by the results of his experiments to make new hypotheses, while he depends, with respect to the compatibility of the new hypotheses with the old axioms, solely upon these experiments or upon a certain physical intuition, a practice which in the rigorously logical building up of a theory is not admissible. The desired proof of the compatibility of all assumptions seems to me also of importance, because the effort to obtain such proof always forces us most effectually to an exact formulation of the axioms.

David Hilbert( Paris 1900)

 

[Organic]

General Information Framework (GIF) set theory


Set (which notated by ‘{‘ and ‘}’) is an object that used as General Information Framework.

Set's property depends on its information’s type.

There are 2 basic types of information that can be examined through GIF.

1) Empty set ={}
2) Non-empty set

(2) has 3 non-empty set’s types:

1) Finitely many objects ( {a,b} ).
2) Infinitely many objects ( {a,b,…} ).
3) Infinite object ( {__} }.

(3) Is the opposite of the Empty set, therefore {__}=Full set.

GIF has two limits:

The lowest limit is {}(=Empty set).

The highest limit is {__}(=Full set).

Both limits are unreachable by (1) or (2) non-empty set’s types.

Infinitely many objects ( {a,b,…} ) cannot be completed, therefore words like ‘all’ or ‘complete’ cannot be used with sets that have infinitely many objects.

{} or {__} are actual infinity.

{a,b,...} is potential infinity.

An example:

http://www.geocities.com/complementarytheory/LIM.pdf

 

[master_coda]

Originally posted by Organic
General Information Framework (GIF) set theory

So what can we do with this theory that we can't do with standard set theory?

 

[phoenixthoth]

In what I have read (long ago) on multivalued logics, the classical paradoxes in binary logic can fairly straightforwardly be extended to multi-valued logic.

For instance, $S := \{ x | x \in x\; \mbox{is not true} \}$ sufficies for at least one ternary logic.

i'm going to have to look at this further. since you replaced it with the word false, i'd just like to say that things are not exclusively true or false anymore but possibly the third truth value. just for everyone else, i'll put what i know about 3 valued logic here:
let 0 mean F, .5 mean the third truth value M, and 1 mean T.
V(P) gives the truth value of the property (aka well formed formula) P. to use "max" below, we could just say F < M < T.

V(AvB)=V(BvA)=max{V(A),V(B)}.
V(~A)=1-V(A).

(in this language, your S is $$S:=\left\{ x\in U:V\left( x\in x\right) \neq T\right\}$$ and i'll think about this. is that something you can state in 1st order logic as the subsets axioms is stated?)

from this, one can derive the truth tables for the other truth values using the rules A^B=~(~A v ~B), A->B = ~A v B, and <-> is what it usually is.

apparently, there are 3072 3 valued logics but i doubt all of those are generalizations of 2 valued logic.

note that V can be any function that generalizes 2valued logic and this one does. let me now write out some truth tables, ending in the truth table for the argument by contradiction, which is $$\left[ A\rightarrow \left( B\leftrightarrow \symbol{126}B\right) \right] \rightarrow \symbol{126}A$$. the first two columns will be the truth values of A and ~A. then there will be a double bar ||. the next truth values will be B and ~B, followed by B->~B and ~B->B. the next pair will be B<->~B and A->(B<->~B). the final truth value will be ~A. the main result is that the final values are not always T; ie, this argument is not applicable because it's not a tautology anymore.
1.TF||TF||FT||FF||T
2.TF||MM||MM||MM||M
3.TF||FT||TF||FF||T
4.MM||TF||FT||FM||M
5.MM||MM||MM||MM||M
6.MM||FT||TF||FM||M
7.FT||TF||FT||FT||T
8.FT||MM||MM||MT||T
9.FT||FT||TF||FT||T

the standard russell's paradox is to use the following:
A states the universal set exists (normally, ~A is T)
B states that S &isin; S.

and there's the rub; we need to know what a "safe" set of formulas is. This is a purely metamathematical concern; I can't see any way it could be written formally.

it would require quantifying over wffs as far as i can see. this would be added somewhere:
if there is a wff such that it implies a contradiction, then there is no subset with that wff as a defining property.
yes, this would involve an investigation of "safe" wffs. we would only have to try this if one were to not allow 3-valued logic to get around what russell's paradox is currently.

there's no sign that some other paradox won't rule U out; if there are others, do they also rely on contradiction which is no longer a tautology in 3 valued logic? I'm not suggesting we use 3 valued logic for everything. it is a generalization of 2 valued logic used only when necessary such as to say things like "from one perspective, russell's theorem is a nontautology."

Anyways, it is an interesting exercise to formally write up your ternary logic and see if it really sufficies. I bet that replacing "$p \notin q$" in Cantor's argument with "$p \in q$ is false or <name of third logical value>", then you can still derive a contradiction.

i'll look at your S. thanks for submitting it.

organic,
i think what I've been doing is what you're after but it only applies to the universal set, not just every infinite set. you said it, in spirit: binary logic cannot handle infinitely many objects. while that's technically incorrect, it is definitely true of the absolute infinity. and i think i can show how 3 valued logic can handle russell's paradox.

for anyone interested, the stanford encyclopedia has nice articles on many-valued logic:
http://plato.stanford.edu/entries/logic-manyvalued/
there it gives the possibility of using it to resolve some paradoxes but it doesn't mention russell's from what i saw.

 

[Organic]

Hi master_coda,

Good question:

Please look at this two articles of mine:

http://www.geocities.com/complementarytheory/ET.pdf

http://www.geocities.com/complementarytheory/CATheory.pdf


At this stage you have to look at them as non-formal overviews, but with a little help from my friends, they are going to be addressed in a rigorous formal way.

All the energy that was used to research the transfinite universes, is going to be used to research the information concept itself, including researches that explore our own cognition's abilities to create and develop the Math language itself.

By GIF set theory our models does not have to be quantified before we can deal with them, because GIF set theory has the ability to deal with any information structure in a direct way, which keeps its dynamic natural complexity during the research.

Concepts like symmetry-degree, Information's clarity-degree, uncertainty and redundancy, are some of the fundamentals of this theory.


Organic

 

[phoenixthoth]

hurkyl,

define S so that $$S=\left\{ x\in U:V\left( x\in x\right) \neq T\right\}$$.

let P be the statement S &isin; S and Q be the statement $$V\left( S\in S\right) \neq T$$.

in the following truth table, I'm using the same rules as above and the first column is the truth value of P, the second column is the truth value of Q, and the third is the truth value of Q<->~Q:
1. T||F||F
2. M||T||M
3. F||T||F

note that it's not always F. in fact, it's M when $$V\left( S\in S\right) = M$$. thus, one possible conclusion is that if "undecidable" means neither true nor false, then it is undecidable whether an undecidable statement implies Q<->~Q.

hence, one can axiomatize it either way. S &isin; S has truth value M does not lead to a contradiction. in other words, it is "unknown" whether S &isin; S, if to know something means to know it is either true or false.

however, i'd like to avoid attaching meaning to M yet. just arguing syntactically, not semantically based on the meaning of T, M, or F. syntactically, this is a valid system and the contradiction is no longer tautologically false. thus, it appears to me that this expanded paradox is not a paradox in 3 valued logic.

 

[Hurkyl]

The thing is, it seems, Cantor's argument proves true this statement:

$$S \in S \Leftrightarrow \neg(S \in S)$$

However, no truth value assignment to $S \in S$ permits this statement to be true.



I made this truth table, but it turns out I didn't need it for this post. However, I'm leaving it for future reference.

$$\begin{array}{c | c | c | c | c | c | c}<br /> A & B & A \vee B & A \wedge B & \neg A & A \Rightarrow B & A \Leftrightarrow B \\<br /> \hline<br /> T & T & T & T & F & T & T \\<br /> T & M & T & M & F & M & M \\<br /> T & F & T & F & F & F & F \\<br /> M & T & T & M & M & T & M \\<br /> M & M & M & M & M & M & M \\<br /> M & F & M & F & M & M & M\\<br /> F & T & T & F & T & T & F \\<br /> F & M & M & F & T & T & M \\<br /> F & F & F & F & T & T & T<br /> \end{array}$$

 

[Hurkyl]

On ternary deduction

It occurs to me that ternary logic has some annoying problems... for instance, $A \Leftrightarrow A$ is not a tautology. It seems one can no longer use $A \Rightarrow B$ to be synonymous with "If A then B", and one can no longer use $A \Leftrightarrow B$ to be synonymous with "A and B are the same"...

 

[phoenixthoth]

thanks hurkyl. would it be possible to just use ternary logic to say that russell's paradox is based on a nontautology, axiomatize the universal set into existence that way, look for other paradoxes that ternary logic can't handle, and then stick to binary logic for the rest of set theory? if one ever uses ternary, are they bound to always use it, kind of a consistency thing from an esthetic viewpoint vs consistency from a strictly mathematical viewpoint?

i would like to point out that A<->A is a tautology if V(A)!=M. since nothing else in set theory requires ternary logic, it seems like nothing is harmed.

i was also thinking of using a lattice-theoretic approach to set theory in which Ø and U are axiomatized into the system, along with the usual other sets. some problems:
1. how to define &isin;
2. how to define subset
3. how to reformulate the subsets axiom
4. avoid russell's paradox and ternary logic.

the thing is, i bet people have tried all this before at some point but since i don't know about it, it probably either all failed or never caught on.

 

[Hurkyl]

I've never seen a non-binary logic presented at a foundational level; I can't say what's a "normal" thing to do.


Anyways, an approach that works for general purpose is to strike out the axiom of the power set; you only work in an explicitly specified "universe" (which is not the universe of all sets). For instance, one can (I believe) do the whole of analysis in a thing called a "superstructure" which is constructed as:

$$\mathcal{S}_i = \left\{<br /> \begin{array}{l l}<br /> \mathbb{R} \quad & i = 0 \\<br /> \mathcal{S}_{i-1} \cup \mathcal{P}(\mathcal{S}_{i-1}) & i > 0<br /> \end{array}<br /> \right$$

and the superstructure is defined to be $\mathcal{S} = \bigcup_{\iota \in \mathbb{N}} \mathcal{S}_{\iota}$.

So in the superstructure, you can indirectly find the power set of many sets by appealing to the structure of the superstructure, but you aren't allowed to take the power set of an arbitrary set.


(note that $\mathcal{S}$ is a lattice, if you use union for join and intersection for meet... or is it the other way around?)

 

[phoenixthoth]

you can further that structure if it's not large enough by replacing R with S.

i always wondered why my set theory teacher told me some set theorists don't like the powerset axiom...

 

[Hurkyl]

Right. (and on a side note, if you replace $\mathbb{R}$ with $\mathbb{N}$, you get the same thing)


But the thing is, it is large enough, so you don't need to do allow anything recursive to occur.


Intuitively, all of the bad things start happening when you allow sets to get "too big". The power set axiom is, I believe, the only way to make big sets in ZFC, so if you get rid of it, things won't get too big!

 

[phoenixthoth]

there is no upper "limit" to this, is there?

more generally, a universal set could never be constructed from union and powerset can it? i don't think it can because the powerset is always bigger. even some kind of transfinite recursion, if such a thing exists, would have to be of a nonabsolute infinitary nature thus not big enough to construct U "from below."

 

[Hurkyl]

Right, the power set is always bigger. If you could prove the existence of a universal set, that would be a contradiction in ZFC.


There is such a thing as transfinite recursion, but it's tricky to set up... I don't see how you could even think about pointing towards a universal set, though; the two biggest problems I see are:

1) ZFC permits sets that cannot be constructed from its axioms.
2) The index "set" would be "too big" to be a set

 

[phoenixthoth]

hurkyl: If you could prove the existence of a universal set, that would be a contradiction in ZFC.

using binary logic. so if i continue working on the assumption that it's ok to use ternary logic once, to climb over the proof by contradiction to show that the universal set can't exist, i would then kind of be a math rebel? rebel or just plain a waste of time? I'm sure someone must have tried this before and i haven't heard of it so it must have failed, right? will you be the dedekind to my cantor? ;)

 

[Hurkyl]

Ok, spend a few years developing ternary logic and set theory, then I'll show you the flaw.

 

[Hurkyl]

But more seriously, you can't just use ternary logic once; if the theory is written in binary logic, you can't introduce ternary logic where you fancy it; you have to either start with ternary logic from the beginning, or use binary logic to model a theory which uses ternary logic.

 

[phoenixthoth]

do you promise? can i show it to you as i develop it instead of after 3 years?

ok, that's something i had been wondering. i have to recheck what few proofs i have and learn how to prove via ternary logic.

 

[Hurkyl]

It would certainly be interesting to look at; I'll give you fair warning, though, I'm lazy so I might not be up to doing a lot of research for you.

 

[phoenixthoth]

what do you think of the following general plan:
revise all axioms so that whenever there is a well-formed formula W appearing in the axiom that is meant to be true, replace it with
V(W)=T.

for example, if we want A<->B to mean what it usually does, we can say V(A<->B)=T instead of just A<->B. however, if V(A) and V(B) are both M, then V(A<->B)!=T, though that might be a very good thing. so, in other words, replacing A<->B with V(A<->B)=T might be just what i need.

in the axiom of extensionality, there's a statement of the form
$$\forall x\left( x\in a\leftrightarrow x\in b\right) \rightarrow a=b$$. in fuzzy logic, these connectives don't mean what they do in binary logic; so let's see what happens when i modify it to this:
$$\forall x\left( V\left( x\in a\leftrightarrow x\in b\right) =T\right) \rightarrow a=b$$.

the subtlety is whether or not binary logic must be used when i make a statement like $$V\left( x\in a\leftrightarrow x\in b\right) =T$$. with equality, i want it to either be = or !=. that doesn't seem to be a problem. equality is binary whereas the truth value of what's inside the V( ) can be T, M, or F. how does all that sound before i continue?

what i'd like to do is show that this axiom is equivalent to the original axiom. i'll have to think about this.

the main sticking point will be the subsets axiom. how about this:
SS 2. $$\exists x\forall yV\left( \left( y\in x\leftrightarrow y\in a\wedge A\left( y\right) \right) \right) =T$$?
this subsets axiom does contradict the universal set axiom.

SS 3. $$\exists x\forall yV\left( \left( y\in x\leftrightarrow y\in a\wedge A\left( y\right) \right) \right) \neq F$$.
SS3 does not contradict the universal axiom and V(S&isin;S)=M follows. thus, S could be called a fuzzy subset of U.

the hope is that if the subsets axiom is the only one with any structural difference between it and the original axiom, changing a well formed formula W into V(W)=T in all cases except the SS axiom, in that case changing it to V(W)!=F, then ternary logic will work well.

 

[Hurkyl]

I think it might be better to start with ternary logic (e.g. what are the rules of deduction... do we have rules of deduction only for T, or for both T and M? that sort of thing), then move onto ternary ZFC... or maybe a simpler theory first.

Doing it in its own thread might be nice too

 

[phoenixthoth]

Infinitely many objects ( {a,b,…} ) cannot be completed, therefore words like ‘all’ or ‘complete’ cannot be used with sets that have infinitely many objects.

i agree that complete can't, but why "all?" the quantifier is as in "all sets". why can we not use this? the "therefore" seems to be a non sequitor: the conclusion doesn't follow from the premise.

saying the empty set is {} is meaningless. you have to define it and give it properties such as for all sets x x is not an element of {}.

saying the (absolutely) infinite set {__} without properties is also meaningless. you have to say that for all sets x x is an element of {}. in your theory, you also have to show how russell's paradox is avoided.

 

[Organic]

Hi phoenixthoth,

Please let's continue here:

https://www.physicsforums.com/showthread.php?s=&threadid=11440


Thank you.


Yours,


Organic

 

[moshek]

Non Euclidian mathematics

Organic,

Well, You are trying to develop here some non Euclidean
mathematics, so exact definition at the beginning is not the point.

but better you read what happande to Hipasus:

http://www.anselm.edu/homepage/dbanach/pyth3.htm

Happy new year
Moshek

 

[Organic]

Hi Moshek,

Can you show some fundamental differences between Euclidean and Non-Euclidean mathematics?

 

[phoenixthoth]

one difference is that the sum of interior angles in a triangle is not 180 degrees.

an axiom of euclidean geometry is that given a line and a point not on the line, there is 1 line passing through the point that is parallel to the given line. in non euclidean geometry, this can be replaced with a number other than 1.

 

[moshek]

It will be very difficult organic because it is so new that it is almost does not exist. Only in the mind of maybe ~10 people in the world today but they are not a community like the big Euclidean mathematics community that exist more that 2000 year's and work in Plato paradigm.

very interesting also is this paper:

http://www.american.edu/academic.depts/cas/mathstat/People/kalman/pdffiles/irrat.pdf


Moshek