The search for absolute infinity

[Hurkyl]

I think that expanding a set's concept to include redundancy yields some concept different from the set concept. In fact, we have a name for it; a multiset.

 

[Organic]

Hi Hurkyl,

My point of view leading me to what I call Complementary logic, which is a fading transition between Boolean logic (0 Xor 1) and non-boolean logic (0 And 1), for example:

Number 4 is fading transition between multiplication 1*4 and addition ((((+1)+1)+1)+1) ,and vice versa.

These fading can be represented as:

(1*4)              ={1,1,1,1} <------------- Maximum symmetry-degree, 
((1*2)+1*2)        ={{1,1},1,1}              Minimum information's clarity-degree
(((+1)+1)+1*2)     ={{{1},1},1,1}            (no uniqueness)
((1*2)+(1*2))      ={{1,1},{1,1}}
(((+1)+1)+(1*2))   ={{{1},1},{1,1}}
(((+1)+1)+((+1)+1))={{{1},1},{{1},1}}
((1*3)+1)          ={{1,1,1},1}
(((1*2)+1)+1)      ={{{1,1},1},1}
((((+1)+1)+1)+1)   ={{{{1},1},1},1} <------ Minimum symmetry-degree,
                                            Maximum information's clarity-degree 
                                            (uniqueness)
============>>>

                Uncertainty
  <-Redundancy->^
    3  3  3  3  |        3  3           3  3
    2  2  2  2  |        2  2           2  2
    1  1  1  1  |  1  1  1  1           1  1     1  1  1  1
   {0, 0, 0, 0} V {0, 0, 0, 0}   {0, 1, 0, 0}   {0, 0, 0, 0}
    .  .  .  .     .  .  .  .     .  .  .  .     .  .  .  .
    |  |  |  |     |  |  |  |     |  |  |  |     |  |  |  |
    |  |  |  |     |__|_ |  |     |__|  |  |     |__|_ |__|_
    |  |  |  |     |     |  |     |     |  |     |     |
    |  |  |  |     |     |  |     |     |  |     |     |
    |  |  |  |     |     |  |     |     |  |     |     |
    |__|__|__|_    |_____|__|_    |_____|__|_    |_____|____
    |              |              |              |

4 =
                                   2  2  2
          1  1                     1  1  1        1  1
   {0, 1, 0, 0}    {0, 1, 0, 1}   {0, 0, 0, 3}   {0, 0, 2, 3}
    .  .  .  .      .  .  .  .     .  .  .  .     .  .  .  .
    |  |  |  |      |  |  |  |     |  |  |  |     |  |  |  |
    |__|  |__|_     |__|  |__|     |  |  |  |     |__|_ |  |
    |     |         |     |        |  |  |  |     |     |  |
    |     |         |     |        |__|__|_ |     |_____|  |
    |     |         |     |        |        |     |        |
    |_____|____     |_____|____    |________|     |________|
    |               |              |              |


   {0, 1, 2, 3}
    .  .  .  .
    |  |  |  |
    |__|  |  |
    |     |  |
    |_____|  |
    |        |
    |________|
    |

Multiplication can be operated only between objects with structural identity .

Also multiplication is noncommutative, for example:

2*3 = ( (1,1),(1,1),(1,1) ) or ( ((1),1),((1),1),((1),1) )

3*2 = ( (1,1,1),(1,1,1) ) or ( ((1,1),1),((1,1),1) ) or ( (((1),1),1),(((1),1),1) )

More about the above you can find here (the first part of it is your definitions):

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

More about Complementary logic, you can find here:

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

http://www.geocities.com/complementarytheory/4BPM.pdf



Organic

 

[phoenixthoth]

seems like !z∈_?z doesn't lead to a paradox but the usual russell's paradox is still a paradox.

 

[master_coda]

Organic,

I think the point that Hurkyl was trying to make was that if you take the concept of a set and add more properties to it, such as redundancy, you aren't really just using a set anymore.

That's perfectly fine, of course. That's why we have objects like ordered sets and multisets.

 

[Hurkyl]

I hate to encourage a hijacking (iow, promote your theory in your thread)... but it seems like this should also be valid (among other things):

2 * 3 = <<1>, 1> * <1, <1, 1>> = <<<1>, 1>, <<<1>, 1>, <<1>, 1>>>

 

[Hurkyl]

Phoenix: I was doing a bit of thinking, and I wonder if you shouldn't just study fuzzy sets:

For instance, consider this model:

We assign real numbers to truth values: T = 1, M = 0.5, F = 0. Your sets are fuzzy sets whose range must be {0, 0.5, 1}. In other words, a set is merely a function from a domain into {0, 0.5, 1}.

We can then equip a fuzzy set with an extra gadget we call it's "default value"; any element that doesn't appear in the domain of the fuzzy set gets assigned the default value.

for instance, consider the fuzzy set over {0, 1, 2, 3}:

S = { (0, T), (1, T), (2, F), (3, M) }

That is, in your set land, 0 and 1 are elements of S, 2 is not an element of S, and 3 is a maybe-element of S. Or, S(0) = 1, S(1) = 1, S(2) = 0, S(3) = 0.5

Now, we add the default value "M" to T to get (I'm inventing notation now):

S = {M | (0, T), (1, T), (2, F), (3, M) }

So now all of the above values are the same, but we also have (and this is technically new notation): 4 is a maybe-element of S; that is S(4) = 0.5, and similarly for everything that's not 0, 1, or 2.


And then, via magic, we have a "universal set" produced by taking the empty set (which is a fuzzy set with the empty domain!) and equipping it with default value T.


We could replace the range of truth values {0, 0.5, 1} with any domain we like, really... it seems there's no need to resort to ternary logic with this approach, we can accomplish the same sort of thing with crisp sets.

We get consistency relative to ZFC for free this way... I wonder how the axiomization of this theory would work?


Allow me to reformulate it in a slightly different way to simplify things.

A p-set is an ordered pair $(S, d)$ where we have $\mathrm{Set}(S)$ and $d \in \{\mathrm{true}, \mathrm{false} \}$, and we define p-membership as $a \in_p (S, d) \leftrightarrow a \in S \oplus d$. (where $\oplus$ is exclusive-or; I didn't want to figure out how to make the usual symbol)


That is, $d$ is a sort of polarity; if $d = \mathrm{true}$, then the "default" value is true, and $S$ contains the elements that aren't a p-member of $(S, d)$, and if $d = \mathrm{false}$, then the "default" value is false, and $S$ contains the elements that are a p-member of $(S, d)$


The universal p-set is then $(\varnothing, \mathrm{true})$.

 

[Organic]

I updated my previous post.

Please look at it.

Thank you.

Hurkyl, I think my two last posts are conncted also to the idea of 3 valued logic, but I will open a new thread for it.



Organic

 

[phoenixthoth]

Originally posted by Hurkyl
Phoenix: I was doing a bit of thinking, and I wonder if you shouldn't just study fuzzy sets:

For instance, consider this model:

We assign real numbers to truth values: T = 1, M = 0.5, F = 0. Your sets are fuzzy sets whose range must be {0, 0.5, 1}. In other words, a set is merely a function from a domain into {0, 0.5, 1}.

We can then equip a fuzzy set with an extra gadget we call it's "default value"; any element that doesn't appear in the domain of the fuzzy set gets assigned the default value.

for instance, consider the fuzzy set over {0, 1, 2, 3}:

S = { (0, T), (1, T), (2, F), (3, M) }

That is, in your set land, 0 and 1 are elements of S, 2 is not an element of S, and 3 is a maybe-element of S. Or, S(0) = 1, S(1) = 1, S(2) = 0, S(3) = 0.5

Now, we add the default value "M" to T to get (I'm inventing notation now):

S = {M | (0, T), (1, T), (2, F), (3, M) }

So now all of the above values are the same, but we also have (and this is technically new notation): 4 is a maybe-element of S; that is S(4) = 0.5, and similarly for everything that's not 0, 1, or 2.


And then, via magic, we have a "universal set" produced by taking the empty set (which is a fuzzy set with the empty domain!) and equipping it with default value T.


We could replace the range of truth values {0, 0.5, 1} with any domain we like, really... it seems there's no need to resort to ternary logic with this approach, we can accomplish the same sort of thing with crisp sets.

We get consistency relative to ZFC for free this way... I wonder how the axiomization of this theory would work?


Allow me to reformulate it in a slightly different way to simplify things.

A p-set is an ordered pair $(S, d)$ where we have $\mathrm{Set}(S)$ and $d \in \{\mathrm{true}, \mathrm{false} \}$, and we define p-membership as $a \in_p (S, d) \leftrightarrow a \in S \oplus d$. (where $\oplus$ is exclusive-or; I didn't want to figure out how to make the usual symbol)


That is, $d$ is a sort of polarity; if $d = \mathrm{true}$, then the "default" value is true, and $S$ contains the elements that aren't a p-member of $(S, d)$, and if $d = \mathrm{false}$, then the "default" value is false, and $S$ contains the elements that are a p-member of $(S, d)$


The universal p-set is then $(\varnothing, \mathrm{true})$.

let me see if i got this right. how would we embed a normal set into its p-set? would a set K be mapped into (K,false)?

 

[Hurkyl]

Right:

$$\forall K \in \mathrm{Set} \forall a: a \in K \Leftrightarrow a \in_p (K, \mathrm{false})$$

 

[phoenixthoth]

something I'm going to think about, too, is whether (Ø,true) violates russell's paradox and whether (K,true) is a set.

 

[master_coda]

Originally posted by Hurkyl
A p-set is an ordered pair $(S, d)$ where we have $\mathrm{Set}(S)$ and $d \in \{\mathrm{true}, \mathrm{false} \}$, and we define p-membership as $a \in_p (S, d) \leftrightarrow a \in S \oplus d$. (where $\oplus$ is exclusive-or; I didn't want to figure out how to make the usual symbol)


That is, $d$ is a sort of polarity; if $d = \mathrm{true}$, then the "default" value is true, and $S$ contains the elements that aren't a p-member of $(S, d)$, and if $d = \mathrm{false}$, then the "default" value is false, and $S$ contains the elements that are a p-member of $(S, d)$


The universal p-set is then $(\varnothing, \mathrm{true})$.

But how many properties of "standard" ZFC sets would also apply to p-sets? Any ZFC set A has an "equivalent" p-set $(A,\mathrm{false})$ but not every p-set has an equivalent ZFC set.

So even though you can create a universal set, we still need to figure out what we can do with it.


PS. I think the LaTeX symbol you were looking for is $\veebar$. I like it better than $\oplus$.

 

[Hurkyl]

Well, you can do union and intersection. I don't know if you can actually do anything interesting with them, though; I'm not entirely sure what Phoenixthoth's goal is in his pursuit.

 

[phoenixthoth]

my ultimate goal is to look at self-aware structures as per that article by max tegmark. if we track all possible configurations of the universe over time we get some kind of set, possibly a manifold. since we are in this set, and we are self aware, it makes sense that some sets and/or logical structures have SA-structure. if U is the set of all possible configurations of the universe, i was wondering if U would have SA or if it just contains things that do. if it does, would it, in a sense, be omniscient of all its contents? I'm not even sure how would would define SAS's. perhaps they'll be left undefined, not unlike the word set, and axioms will be given that govern them.

for instance... if x is a set with SAS and x can be bijected to y, then y also has SAS.

however, if SAS has to do with more than number of elements but something more complicated, then isomorphic (wrt some operation(s)) would be more appropriate. the thing is that static sets can't prove to us they are self-aware or even express that fact. a changing three dimensional manifold becomes a static 4D manifold. what's changing is our awareness of it which some to be a concept not bound by time. well, this is way off course now...

would there be different kinds of SA-structure? and a way to measure SA-structure?

i have a thread with random conjectures here:
http://207.70.190.98/scgi-bin/ikonboard.cgi?;act=ST;f=2;t=196

back to a universal set... my question was the same as master_coda's, which was whether (K,true) always corresponds to a set.

seems like (Ø,true) is just like {x|x=x} which is a proper class.

 

[Hurkyl]

Right. The p-membership of $(S, \mathrm{true})$ is always a proper class for any set $S$.

The major drawback of p-sets, as I've defined them, seems to be that it cannot be the case that both the p-membership and p-nonmembership are both proper classes.

However, my gut says it's at least subtheory of your goal, and would be worth studying.

 

[phoenixthoth]

another attempt

second truth table (REPEAT):

$$\begin{array}{cccccccc}<br /> A & B & A\rightarrow B & A\rightarrow _{T}B & A\leftrightarrow B & <br /> A\leftrightarrow _{T}B & A\leftrightarrow _{=}B & A\leftrightarrow _{\neq F}B<br /> \\ <br /> T & T & T & T & T & T & T & T \\ <br /> T & M & M & F & M & F & F & T \\ <br /> T & F & F & F & F & F & F & F \\ <br /> M & T & T & T & M & F & F & T \\ <br /> M & M & M & F & M & F & T & T \\ <br /> M & F & M & F & M & F & F & T \\ <br /> F & T & T & T & F & F & F & F \\ <br /> F & M & T & T & M & F & F & T \\ <br /> F & F & T & T & T & T & T & T<br /> \end{array}$$
define a new conditional:
$$\begin{array}{ccccc}<br /> A & B & A\rightarrow _{+}B & A\leftrightarrow _{+}B & A\leftrightarrow _{+}<br /> \symbol{126}A \\ <br /> T & T & T & T & F \\ <br /> T & M & T & T & F \\ <br /> T & F & F & F & F \\ <br /> M & T & T & T & T \\ <br /> M & M & T & T & T \\ <br /> M & F & T & T & T \\ <br /> F & T & T & F & F \\ <br /> F & M & T & T & F \\ <br /> F & F & T & T & F<br /> \end{array}$$

the axioms of subsets would read this:

axiom of subsets version 6:

$$\exists x\forall y\left( y\in _{?}x\leftrightarrow _{+}y\in<br /> _{?}a\wedge A\left( y\right) \right)$$.



if there is a universal set $$U$$, and we let $$a=U$$ and $$A\left( y\right) =y\notin y$$, note that $$y\notin y$$ is a binary statement and $$y\in _{?}x$$ is a ternary statement. here's how russell's paradox, which was a problem for
those other biconditionals, would work:

$$\begin{array}{ccc}<br /> S\in _{?}S & S\notin S & S\in _{?}S\leftrightarrow _{+}S\notin S \\ <br /> T & F & F \\ <br /> M & F & T \\ <br /> F & T & F<br /> \end{array}$$

axiom of foundation states this:

$$\forall a\left[ a\neq \emptyset \rightarrow \exists x\in a\left( x\cap a=\emptyset \right) \right]$$. if you consider $$\left\{ a\right\}$$, you can show that for no ZFC sets is the following true: $$a\in a$$. this axiom is inconsistent with the universal axiom because $$U\in U$$. (note that $$S\in S$$ is false because $$S\in _{M}S$$.) perhaps a way to restate the universal set axiom is this:

$$\exists !x\left( x\in x\right)$$. one then has to modify the foundation axiom to this:

$$\forall a\left[ a\neq \emptyset \wedge a\neq \left\{ U\right\} \rightarrow \exists x\in a\left( x\cap a=\emptyset \right) \right]$$ or some such. instead of {U}, it would have to be any set containing U as an element, i think.

 

[phoenixthoth]

if $$V\left( y\in _{?}a\wedge A\left( y\right) \right) =M$$ then that gives no information on xRy where R is $$\in$$ or $$\in _{M}$$ or $$\notin$$.

i'm finding it difficult to find any fuzzy sets besides the set S in russell's paradox which may be a good thing.

 

[phoenixthoth]

summary of current work

all feedback appreciated. there should be a file attached which is a zipped pdf.

 

[Hurkyl]

I haven't tried to sit down and digest all of it yet... I've recently decided to take up the task if trying to fully digest the axiom of foundation; i.e. how to prove:

$$\forall S \forall x \in S : S \notin x$$
$$\forall S \forall x \in S \forall y \in x: S \notin y$$

et cetera (including transfinite membership chains)


As one of the PDFs you linked mentions, you don't necessarily need to have the axiom of foundation in your set theory anyways.


I thought a bit more about ternary logic, and I think (though I may be wrong) that it should be approached in this way:

Use binary logic simulating ternary logic; e.g. one might have the binary function Q(P) which simulates the ternary fact V(P) = M.

(Incidentally, it seems that you are using this approach, whether consciously or not)

If not that, I'm beginning to think that ternary deduction might be richer than ternary truth tables, and that making truth tables isn't the right way to approach making deductions.

 

[Hurkyl]

e.g. I can prove the first of these:

Assume $x \in S \wedge S \in x$. Consider $T := \{S, x\}$

By the axiom of foundation, one of $S \cap T$ and $x \cap T$ must be the null set.

However, $S \cap T = \{x\}$, and $x \cap T = \{S\}$, which is a contradiction.

 

[phoenixthoth]

Originally posted by Hurkyl
e.g. I can prove the first of these:

Assume $x \in S \wedge S \in x$. Consider $T := \{S, x\}$

By the axiom of foundation, one of $S \cap T$ and $x \cap T$ must be the null set.

However, $S \cap T = \{x\}$, and $x \cap T = \{S\}$, which is a contradiction.

F2, foundation 2, is
$$\forall a\left( \left( a\neq \emptyset \wedge U\notin a\right) \rightarrow \exists x\in a\left( x\cap a=\emptyset \right) \right)$$

this would mean that your T (a in the axiom) wouldn't be an applicable use of F2 (edit: if S or x was U) though it would be of the usual axiom of foundation. otherwise if U is not one of them, you can't have a chain of memberships that start and end with the same set i don't think.

but if this proves to be a problem, i'll have to see what it would unravel to loose foundation. I'm only partway through an article on set theory and he said foundation will have bigger implications but for now, it is used to show that no set is an element of itself (which is why i needed to change it to allow for U in a way that if U weren't around it would be the same axiom).

the main boon in this system is that most things remain crisp with the use of the new membership symbols so i can still use deduction and contradiction. there was another article where someone almost did what i did, by computer scientists no less, in which they invented two new membership symbols. i emailed one of them my ideas but never got a reply. they were really close to turning their guns on russell's paradox but i think that's been considered a dead horse for like 150 years now. anyway, with the crispiness of things mostly intact, except with the subsets axiom, I'm hoping that everything will turn out all right in the end.

in fact, i can't find many fuzzy sets in this theory. for example, I'm not sure i wouldn't have to modify the pairing axiom in order to get a set like x={(Ø,M), ({Ø},T)} where that means that the truth value of Ø&isin;x is M and the truth value of {Ø}&isin;x is T, using the old membership symbols here. in the new system, x&isin;y, x&isin;_My, and $$x\notin y$$ are mutually exlusive.

i'm glad i didn't call this thread naked ladies! thank you very much for all your feedback.

edit: if you would, please expound on what you mean by ternary deduction without truth tables. i think that would be a great boon to tuzfc (ternary universal zfc) though i only have an incling for what that would entail (pun intended). i don't know if I'm jumping the gun here, but, to be conservative, let's say that in 3 years there is a publishable paper in this; well, it could be a joint paper with tuzfc and ternary deduction combined somehow.

if this works out, i would like to say the cardinal number of U is capital omega or capital alpha, alternatively. also, what is the standard letter for the third truth value? is it M? one could be kind of ridiculous and call this m-theory but rather than be ambiguous about what this M stands for, i'll say it stands for 'maybe' and the eastern concept called mu which is actually why i called it M. in truth, this eastern approach is what proved to be a major inspiration to the resolution of certain paradoxes not unlike how it 'resolves' certain koans.

 

[Organic]

What are the results of http://mathworld.wolfram.com/Indeterminate.html
when using your theory.

 

[phoenixthoth]

an odd result that makes sense

i have more for you to digest once the inital part has cleared.

an example:

if $$x$$ is a proper subset of $$U$$, then there is no map from $$x$$ onto $$U$$.

this entails that for all sets $$a$$, $$U\backslash \left\{ a\right\}$$ is not in bijection with $$U$$ and $$U\backslash \left\{ a\right\} \neq U$$ which is a divergence from the usual case of infinite sets where if $$x$$ is an infinite set then $$x\backslash \left\{ a\right\}$$ is in bijection with $$x$$ and, more generally, if $$a$$ is a set of inferior cardinal number than $$x$$,then $$x$$ is in bijection with $$x\backslash a$$.

so perhaps this can be a precise formulation of the difference between potential infinity, actual infinity, and finity.

this makes sense because you wouldn't expect that if you remove something from U that it should be like U. one can show that if anything is like U, ie in bijection with it, then it is U.

edit: one can also show that all proper subsets of U have strictly less cardinality.

i envision a bottom and a top and the real inaccessible cardinals are the ones in between.

 

[phoenixthoth]

organic,
i think that when one performs those operations with Ø and U, the answer will be indeterminate in some cases. the multiplication would refer to cardinality multiplication. the difference to set difference and the division could refer to another thread i started on division. exponentiation refers to cardinal exponentiation.

edit:
another result: if there is a map from x onto P(x) then either x=U or x is fuzzy.

thus, the only sets for which x might be mapped onto P(x), besides U, are necessarily fuzzy sets. since so few fuzzy sets seem to exist without more axioms, this is a good thing and intuition is not being violated too terribly yet.

i can also show exactly where cantor's diagonal argument fails in general.

 

[Hurkyl]

All right, beginning at the top.

Since deduction in ternary logic is pretty much essential to the whole thing, it's definitely worth spending a bit of time on just that. I've thought more about it, and I understand how it works now.


Recall in binary logic, we might write a deduction (like modus ponens)

P&rarr;Q
P
-----
Q


Since there are only two truth values, this notation is sufficient... we need a way to indicate truth value in ternary logic in a deduction. I think the V(P)=T notation isn't bad for formulas, but I think a different notation is clearer for deduction. Consider modus ponens in your ternary logic:

T: P&rarr;Q
M,T: P
--------
T: Q

This notation means that we know V(P&rarr;Q) = T, V(P) in {M, T}, and we deduce V(Q)=T.

This is actually a valid deduction; if we check the truth table, if V(P&rarr;Q)=T and V(P) in {M, T}, then we must have V(Q) = T! Modus ponens is still useful in your ternary logic!

Also, we have:

M: P&rarr;Q
T: P
--------
M: Q

or more generally, the deduction schema

X: P&rarr;Q
T: P
--------
X: Q

where X ranges through {F, M, T}.

And other familiar rules, like:

T: P&rarr;Q
T: Q&rarr;R
--------
T: P&rarr;R

M: P&rarr;Q
M: Q&rarr;R
--------
M: P&rarr;R

T: P&rarr;Q
M: Q&rarr;R
--------
M,T: P&rarr;R

(For this one, we can only conclude that P&rarr;R is not false)

X: A&rarr;~A
--------
X: ~A

T: A&rarr;F
--------
F: A

T: P&rarr;Q
F,M: Q
--------
F: P

And so on; all of our favorites (I think) have a version or two valid for ternary deduction.


More as I think about it.

 

[Hurkyl]

Hrm...

I've found something disturbing about your axiom of subsets:

$$\forall a \exists x \forall y:<br /> (y \in_? x \leftrightarrow_+ (y \in_? a \wedge A(y)))$$

Suppose we have said set $x$, and we know that, for a particular $y$, $V(y \in_? a)=T$, and $A(y) = T$...

We cannot conclude that $V(y \in_? x)=T$! The axiom is still satisfied if $V(y \in_? x)=M$... we've obliterated crisp ZF.

 

[Hurkyl]

I have an idea for additional predicates. The great thing about binary deduction is that any deduction can be entirely encoded into a formula via $\wedge$ and $\rightarrow$; something similar is needed for ternary logic.


Let's start with the simplest; suppose we were able to prove:

T: A
------
...
------
T: B

We want to write this as a formula F to simplify further proofs, so we can apply the deduction

T: A
T: F
-----
T: B

in one step.


Here's a truth table:

$$\begin{array}{c|c|c}<br /> A & B & A \circ B \\<br /> \hline<br /> T & T & T \\<br /> T & M & X \\<br /> T & F & X \\<br /> M & T & T \\<br /> M & M & T \\<br /> M & F & T \\<br /> F & T & T \\<br /> F & M & T \\<br /> F & F & T \\<br /> \end{array}$$

Where X could be either F or M; at the moment they don't matter. As you can tell, I haven't decided on a symbol for this operation either.

Does this do what we want? If we're given that $V(A \circ B) = T$, then if $V(A)=T$ we can then conclude $V(A)=F$. However, if $V(A) \in \{F, M\}$, we can conclude nothing about $V(B)$.

Example!

I mentioned earlier that:

$$\begin{array}{ll}<br /> T: & A \rightarrow \neg A \\<br /> \hline<br /> T: & \neg A<br /> \end{array}$$

was a valid deduction. Let's try out our new operation:

$$\begin{array}{c|c|c|c}<br /> A & \neg A & A \rightarrow \neg A & (A \rightarrow \neg A) \circ \neg A \\<br /> \hline<br /> T & F & F & T \\<br /> M & M & M & T \\<br /> F & T & T & T \\<br /> \end{array}$$

Joy! $(A \rightarrow \neg A) \circ \neg A$ is a tautology, as advertised!

And, as we can see from the truthtable, if we were given $(A \rightarrow \neg A) \circ \neg A$ and $A \rightarrow \neg A$ were both true, we can immediately conclude $\neg A$ is true, without having to do the (trivial) work to prove it again.

For fun, consider the truth table for $(A \circ B) \wedge (B \circ A)$:

$$\begin{array}{c|c|c|c|c}<br /> A & B & A \circ B & B \circ A & (A \circ B) \wedge (B \circ A) \\<br /> \hline<br /> T & T & T & T & T \\<br /> T & M & X & T & X \\<br /> T & F & X & T & X \\<br /> M & T & T & X & X \\<br /> M & M & T & T & T \\<br /> M & F & T & T & T \\<br /> F & T & T & X & X \\<br /> F & M & T & T & T \\<br /> F & F & T & T & T \\<br /> \end{array}$$

This gives an "if and only if"-like thing; we have the following deductions:

$$\begin{array}{ll}<br /> T: & (A \circ B) \wedge (B \circ A) \\<br /> T: & A \\<br /> \hline<br /> T: & B<br /> \end{array}$$

$$\begin{array}{ll}<br /> T: & (A \circ B) \wedge (B \circ A) \\<br /> F,M: & A \\<br /> \hline<br /> F,M: & B<br /> \end{array}$$

But while $\circ$ is useful as a replacement for $\rightarrow$, I'm not so sure that this thing is useful as a replacement for $\leftrightarrow$.

 

[phoenixthoth]

Originally posted by Hurkyl
Hrm...

I've found something disturbing about your axiom of subsets:

$$\forall a \exists x \forall y:<br /> (y \in_? x \leftrightarrow_+ (y \in_? a \wedge A(y)))$$

Suppose we have said set $x$, and we know that, for a particular $y$, $V(y \in_? a)=T$, and $A(y) = T$...

We cannot conclude that $V(y \in_? x)=T$! The axiom is still satisfied if $V(y \in_? x)=M$... we've obliterated crisp ZF.

i'm proud to have single-handedly obliterated ZF! actually, i do know what you mean and that's a bad thing.

i agree with this and I'm kicking myself for not noticing it. ok. let me run this by you. there are approximately three places i'd like to change the truth table of the plus-biconditional. for one thing, it means it's no longer based on the plus-conditional with the "and" operation, which really isn't serious. my ad hoc justification is this:
all i want to do is find a model for the absolute infinity that doesn't contradict ZF and hopefully not ZFC and any way to do this should be equivalent. i believe this error can be corrected in the following way.

the following letters in a row represent A and B and then the value of the A<->₊B. the A and B refer to what's on the left and right hand side of the subsets axiom.
(optional) changing TMT to TMF. in the subsets axiom 2, when i state the plus conditional, it is assumed as per tradition that it has truth value T. hence if the right hand side has value M then the left hand side is not T.
forsure: changing MTT to MTF. if the right hand side in subsets 2 is T then the left hand side must now be T also. however, if the right hand side is F, then the left hand side is X where X is in {M,F}. i don't know if this is a bad thing.
(optional): changing FMT to FMF.

i need to have this: MFT and this: MMT to avoid russell's paradox.

the new truth table would then be this:$$\begin{array}{ccc}<br /> y\in _{?}x & y\in _{?}a\wedge A\left( y\right) & y\in _{?}x\leftrightarrow_{+}y\in _{?}a\wedge A\left( y\right) \\ <br /> T & T & T \\ <br /> T & M & F \\ <br /> T & F & F \\ <br /> M & T & F \\ <br /> M & M & T \\ <br /> M & F & T \\ <br /> F & T & F \\ <br /> F & M & F \\ <br /> F & F & T<br /> \end{array}$$

now it is unambiguous except in one case. if B is true and the plus-biconditional is true, then A is true. if B is M and the plus-biconditional is true, then A is M. if B is false and the plus-biconditional is true, then A is false or M. this last one bothers me. it means that every set has an indeterminate non-true membership in every other set. but this ain't that bad. or is it?

another thing i was thinking is of cooking up a deep-fat fryer: a "function" like the powerset operation that sends a set to a set of all elements that are in it fully; thus the potentially fuzzy set gets crispy. so if V(y&isin;_?x)=M, and i leave the plus-biconditional, which is now a terrible notation, alone, y will not be in the cooked/fried version of x yet it would M'ly be in x. this is an aside and a diversion for now, i think. the notation in set theory would then really apply to all cooked versions of sets not unlike how in some integration theories (L^p spaces) a function is written but it secretly means the equivalence class of all functions who differ on a set of measure 0.

 

[PFanalog57]

http://en.wikipedia.org/wiki/Singleton_set

In mathematics , a singleton is a set with exactly one element. For example, the set {0} is a singleton. Note that a set such as {{1,2,3}} is also a singleton: the only element is a set (which itself is however not a singleton).
A set is a singleton if and only if its cardinality is 1. In the set-theoretic construction of the natural numbers the number 1 is defined as the singleton {0}.

Structures built on singletons often serve as terminal objects or zero objects of various categories
The statement above shows that every singleton S is a terminal object in the category of sets and functions. No other sets are terminal in that category.
Any singleton can be turned into a topological space in just one way (all subsets are open). These singleton topological spaces are terminal objects in the category of topological spaces and continuous functions. No other spaces are terminal in that category.
Any singleton can be turned into a group in just one way (the unique element serving as identity element These singleton groups are zero objects in the category of groups and group homomorphisms . No other groups are terminal in that category.

The problem with set theory appears to be the primitive concept of : inside / outside

A or not-A

Set theory is based on 2-valued logic of course. Since the absolute infinite is the power set of everything that exists, ergo, an element cannot be removed from it by definition, hence the talk of the removal of elements becomes wishful thinking. A fruitless exercise in futility, for the imagination. Also, the problem arises where, if, an element such as a singleton, is removed from the absolutely infinite, and it[absolute infinity] somehow becomes less than its previous absolute condition, that seriously implies that the absolute infinity is totally non-replenishable, but, actually, if the absolute is generalized as the Platonic state of affairs known as total nothingness, then it no longer has this unprecidented weakness inherited from ZF set theory which is dooomed to die a slow and agonizing death, as die hard academians continue to flog the dead ZF "horse" for all it's worth.



An "excellent idea":

http://www.mmsysgrp.com/stefanik.htm

Thesis: Scientific objectivity is best characterized by the concept of invariance as explicated in category theory than the concept of truth as explicated in mathematical logic.

If reality is self referential, it observes itself, through particle interaction AND conscious observers?

 

[Hurkyl]

The place to put forth your own theories is not in someone else's thread.

 

[PFanalog57]

My apologies Hurkyl, please continue with your interesting discussion.