Boolean Logic cannot deal with infinitely many objects

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The discussion centers on the limitations of Boolean logic and Cantor's diagonalization method when applied to infinite sets. The author argues that Cantor's method cannot account for all combinations in an infinite list, leading to the conclusion that 2^aleph0 equals aleph0, which contradicts established mathematical principles. They present examples of finite combinations and assert that the diagonalization results do not yield new combinations, implying that Boolean logic fails with infinitely many objects. The conversation includes critiques of the author's reasoning and calls for a better understanding of mathematical concepts, particularly Cantor's argument. Ultimately, the thread highlights the complexities and misunderstandings surrounding infinite sets and their treatment in mathematics.
  • #151
to get A as an Empty set we have to define it like that:

COND='all x' Or COND='any x'

if A is a set such that for COND,x not in A, then A={}(=Empty set) .


Do you agree with both COND?

If you're doing literal text substitution, I agree with the statement:

"If A is a set such that for COND, x not in A, then A = {}".

I would like to remark that "all x" is not a grammatical unit, and "for all x" is a quantifier, not a condition.


Emptiness=

As is used throughout mathematics, "=" is a binary relation; it is nonsensical to use it in this way. In order for this statement to have meaning, you are going to have to define it through some means. (axiomatically is fine)


Any x=Emptiness

Similarly, as I mentioned before, "Any x" as used in ordinary logic cannot be used in this way. You are going to have to define this usage through some means as well.
 
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  • #153
In the x computer-model x is like a temporary container that delivers its content to the final destination, which is some set.

You're going to have to define this in some way as well.
 
  • #154
Hurkyl,

http://trochim.human.cornell.edu/kb/dedind.htm

As i see it "all x" is a deductive point of view,

and "any x" is an inductive poit of view.


About the x-container please look here:

http://mathworld.wolfram.com/DummyVariable.html

About the dummy-variable as the x-container.
 
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  • #155
All math is deductive. Logic is deductive if your conclusions follow necessarily from your premises.

Just saying "x is a dummy variable" doesn't explain what x is.
 
  • #156
However, the universal quantifier "for all" has a well-defined meaning.
 
  • #157
hi organic,
you asked us over and over to read your first post and write detailed remarks on it. can you tell us what the flaw in cantor's diagonal argument is to the best of your ability?

theorem: cantor's diagonal argument (one version)
there is no function from any set onto its powerset.

proof: let x be a set. p(x) is its powerset. let f be any function from x to p(x). we will show that f is not onto. to do this requires an element of p(x) (ie a subset of x) that is not mapped to by f. let wf be the well formed fomula (which necessarily depends on the nature of f) that states that "x' is not an element of f(x')", ie, x^{\prime }\notin f\left( x^{\prime }\right). by the axiom of subsets, because wf is a well formed formula, the following object i will define is a set that necessarily depends on the nature of f:

D_{f}:=\left\{ x^{\prime }\in x:w_{f}\left( x^{\prime }\right) \right\} =\left\{ x^{\prime }\in x:x^{\prime }\notin f\left( x^{\prime }\right) \right\}.

i claim that this is an element of p(x) not mapped to by f, which satisfies the requirement and will complete the proof.

to do this, i will argue from \left[ P\rightarrow \left( Q\leftrightarrow \symbol{126}Q\right) \right] \rightarrow \symbol{126}P being a tautology where P and Q are well formed formulae; ie, i will argue by contradiction.

first of all, Df is an element of p(x) because it is a subset, by the subsets axiom, of x.

let P be the statement "f maps an element to Df." in our language, that means \exists x^{*}\in x\left( f\left( x^{*}\right) =D_{f}\right).

to prove ~P, all i must do, because of the quoted tautology, is prove Q<->~Q assuming P is true. suppose \exists x^{*}\in x\left( f\left( x^{*}\right) =D_{f}\right). let Q be the statement "the x*, that exists by assuming P, is an element of Df," i.e., x^{*}\in D_{f}. now we show Q->~Q and then ~Q->Q.

Q->~Q: suppose x^{*}\in D_{f}. remember that f\left( x^{*}\right) =D_{f}, so x^{*}\in f\left( x^{*}\right). but since x^{*}\in D_{f}, this by definition means that x^{*}\notin f\left( x^{*}\right), which the same as ~Q since f\left( x^{*}\right) =D_{f}.

similarly, ~Q implies Q.

QED

comments:
0. i have examined russell's paradox, which involves the same tautology, and cantor's diagonal argument, and i believe i have come up with something.
1. the tautology i quoted is no longer a tautology in fuzzy logic. maybe that's what drove the author royden to intuit that one should eschew all proofs by contradiction...
2. the subsets axiom could be a bad axiom: perhaps sets that lead to contradictions don't exist for consideration.
3. your arguments do not prove that |P(N)|=|N|. however, if you modify the details a bit, you can prove that if U is the absolutely infinite set then not only are P(U) and U in bijection, they are equal! those arguments simply don't work for anything less than U.

read the logic sections under philopsophy in the threads entitled:
1. russell's paradox, the achilles heal of solipsism
2. a new [sic] kind of logic.

please read those before replying.

cheers,
phoenix
 
  • #158
Very well done

Dear phoenix,

Your representation of Cantor theorem is a very well done work.
I am glade that you can see a little similarity to Russell paradox.

I am waiting to Organic replay.


Moshek
 
  • #160
The fact that you start off https://www.physicsforums.com/showth...60549#post60549
by asserting that 0 is a positive number does not speak well for the rest!

Your argument is based on:
This is not logical but a structural point of view on this paradox, and it is based on the simple fact that there can not be any separation between a set and the properties of its contents.

which is non-sense. You seem to interpret this as meaning "a set must have the same properties as its contents" since you use it to assert that the "set of all sets that do not contain themselves" does not contain itself.

Of course, it's not true that "a set must have the same properties as its contents". The set of all positive integers is not a postive integer! The contents of the empty set have the property that they do not exist. Do you assert that the empty set does not exist?
 
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  • #161
Organic Hi,

Phoenix read many of your files. Don't sent him now to read more.

It is now your turn to read his very nice representations
to Cantor theorem and tell us what is wrong there, if at all.

Moshek
 
  • #162
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  • #164
organic,
and we have a proof saying that Boolean Logic cannot
deal with infinitely many objects, in infinitely many magnitudes.


you are correct if you replace "infinitely many objects" with U, the universal set of absolute infinity.

also look up inaccessible cardinals and such things for other objects that are also "really big" that are hard to prove exist as far as i know.

we're really thinking about the same thing but boolean logic can prove that there is no map from x onto p(x) EXCEPT for x=U, the universal set which not only is in bijection with p(U), U=p(U), which is a stronger statement than you say about N, which if i remember right, was something like |N|=|P(N)|. i am about to read your russell's paradox paper and i bet I'm going to find things that are on the right track as well.

ever heard of the hundreth monkey syndrome? it corresponds to the conjecture that often "discoveries" happen independently simultaneously. what wasn't clear to me when i first read your combinations article was that your arguments make more sense if whenever you wrote about N, you were actually writing about the universal set U.

we can say that U can exist axiomatically from the perspective of three valued logic and then stick to two valued logic for the rest in an effort to use two valued logic as much as possible. but your statement that i quoted says this exact thing: boolean logic cannot handle U. kudos to you, organic!

All those kinds of questions are meaningless questions, and they do not lead to any
paradox.

that's exactly what it could mean when a logic statement has the third truth value. this is when mu is the answer. now if you look carefully, you see that it's not U that has meaningless statements surrounding it, it's russell's subset that has meaningless questions surrounding it. i'd also like to point out that cantor's diagonal argument that in 2-valued logic proves that for x!=U, there is no function from x to p(x), and the Df referred to is precisely the set you get in russell's paradox when you consider the identity function from x to p(x) (i think). so the same resolution of the paradox, the appeal to 3 valued logic, applies to show that U is in bijection with p(U). later, i showed that any set in bijection with U is U, hence U=p(U). in fact, any time there is a 1-1 function from U to x, then U=x.

check out the scattered remains on my discussion forum for the search for absolute infinity:
http://207.70.190.98/scgi-bin/ikonboard.cgi?;act=ST;f=2;t=129;st=20;r=1;&#entry573
we were really on the same hunt with combinations and this search.

if only cantor considered using 3 valued logic, we wouldn't be discussing this right now, most likely. his "crippling" attachment to 2 valued logic along with his unwillingness to see past the paradox can definitely drive one mad. I'm not saying we now move to 3 valued logic. all i suggest is that 3valued logic implies the universal set can be axiomatized into existence. furthermore, if that article by max tegmark on the theory of everything is correct, and there are self-aware structures, i postulate that not only is U such a self-aware structure (by no means the smallest SAS), that it, in some weak sense, is omniscient at least of all SAS's that are sets under the supposition that all sets are aware at least weakly of their contents as well as some form of awareness of all sets with nonempty intersection with them and perhaps something also to do with sets they can be mapped onto. I'm just taking a shot in the dark, but I'm guessing that the level of self awareness is in some relation to it's cardinal number. well, if that's true, then U would be the most aware set. however, in that sense, categories would probably be more self-aware and the category of all categories might reign supreme in the self-awareness book. i really don't know what self-awareness is nor how long it will take to figure out what makes us self-aware but for now perhaps if we postulate that we get self-awareness somehow from being in U (note that any manifold such as the one our physical universe is in must be a subset of U), which if any sets have self-awareness implies perhaps that U does, too.

i can't stress highly enough that 2 valued, boolean, logic is just fine for sets that aren't weird subsets of U or U itself.

conjectures on SAS's: http://207.70.190.98/scgi-bin/ikonboard.cgi?act=ST&f=2&t=196&st=&&#entry574
 
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  • #165
HallsofIvy,

To get A as an Empty set we have to define it like that:

x is something

if A is a set such that for any x,x not in A, then A={}(=Empty set) .

But:

x is nothing

if A is a set such that for any x,x not in A, then A=a Non-empty set .

As you can see the property of A depends on the property of x.

The set of all positive integers is not a postive integer

We cannot use the word 'all' together with 'infinitely many objects'
because the basic property of a set with 'infinitely many objects'is not to be completed.

We can use the words 'all' or 'complete' only if we can reach all of their objects, which means, only with sets that have 'finitely many objects'.

This is a fundamental change in the meaning of infinity in Math language.

Please read this:

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

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



Organic
 
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  • #166
To get A as an Empty set we have to define it like that:

x is something

if A is a set such that for any x,x not in A, then A={}(=Empty set) .

But:

x is nothing
correction: x is nothing in particular.

As you can see the property of A depends on the property of x.
no, it doesn't. what properties does it depend on?



We cannot use the word 'all' together with 'infinitely many objects'
because the basic property of a set with 'infinitely many objects'is not to be completed.
the universal set is completed. what you're saying works for U but not for N.

We can use the words 'all' or 'complete' only if we can reach all of their objects, which means, only with sets that have 'finitely many objects'.

This is a fundamental change in the meaning of infinity in Math language.
yes, it is. |U|=OMEGA

on my alephnull website, i proved that U=P(U), remember, so that U really does have the largest cardinal.
 
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  • #167
Hi phoenixthoth,


I am going to read your post (the one with the web sites) and only then I'll reply to you.

Thank you for your detailed reply.


Yours,


Organic
 
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  • #168
i'm waiting to also see what hurkyl thinks.
 
  • #169
correction: x is nothing in particular.
'nothing in particular' is the same as if i say that 'x is something' XOR 'x is nothing'.


Therefore A depends on the property of x.

The axiom of the empty(XOR non-empty) set:

if A is a set such that for any x,x not in A, then A=(depends on the property of x, which can be at least something XOR nothing).


Organic
 
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  • #170
fine. then it is something, but nothing in particular*. however, it does not depend on properties of x. what properties does it depend on? what are you really talking about? Ø, U, or something else? are you saying that it's all basically meaningless? well, from one perspective, it is. they're just symbols. even {} is a meaningless symbol and to lots of people, OMEGA is a meaningless symbol. but if you realize what others have realized, then you can see the meaning.

*duality
 
  • #172
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)
 
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  • #173
If "X = nothing", does that mean X \in \varnothing, X = \varnothing, or something else?

Is it the case that \forall a : a \in \{\_\}?
 
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  • #174
x is the property of some set.

The most basic properties are: Empty XOR Non-empty set.
 
  • #175
Organic,

It appears that even after 15 pages of patient guidance, you continue to use your own ill-formed, imprecise, and ambiguous notation, such as "fullness = {___}" as if no one had ever objected to it.

Haven't you learned anything from these people?

- Warren
 
  • #176
Is "Nothing" is a logical predicate? I.E. "X is nothing iff X = \varnothing"?


Is it the case that \forall a : a \in \{\_\}?
 
  • #178
Hurkyl,

0) There is only one form of nothing, which is the "content" of the Empty set ( {} ).

There are at least three forms of something:

1) finitly many objects ( {a,b} ).

2) infinitely many objects ( {a,b,c,...} ).

3) Fullness ( {__} ).


(0) and (3) are Actual infinity (the lowest and highest limits of any information system).

(2) is potential infinity (cannot be completed).

Please read this:

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

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

Thank you.


Organic
 
  • #179
Does "content" have anything to do with \in?

Is it the case that \forall a : a \in \{\_\}?
 
  • #180
Content is what gives to some set its property(ies).
 

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