Cont: Deeper than primes - Continuation 2

[Nonpareil]

"is a successor of" or "is a member of" are both primitive relations that are not definable in more basic terms.

Sorry, no. The idea of sets themselves is often taken as primitive, but "is a member of" is a very clearly-defined idea: the element in question belongs to the set. It is simple, but fully defined. All relationships are. They have to be, in fact, or trying to use them to describe anything is useless.

If your notion of "successor" is "it's related to its predecessor, but I cannot define how", it is worthless in every respect.

EDIT: And I am deliberately ignoring the rest of your nonsense regarding your nonsensical attempts to redefine infinity by way of my avatar. I don't care. It's been dealt with already, and you repeating the same nonsense interests no one.

 

[jsfisher]

"is a successor of" or "is a member of" are both primitive relations that are not definable in more basic terms.

Set membership has a clearly understood conceptual basis, and while set and set membership are treated as set theory primitives lacking formal definition, their behaviors are full described (effectively, defined) by the axioms.

You are free to provide the same basis for your successor relation, but you cannot simply jump ahead to use it as if it had meaning before you have given it any. As I said, you are free to do that, but that would require new and/or different axioms for your set theory than what ZFC provides.

That still leaves us with you rejecting the Axiom of Infinity. If you'd like to use a modified version of that axiom involving your successor relation, then you are still left with the task of giving it meaning. You may either explicitly define it (as in "for all X, for all Y..."), or you can implicitly define it with new and exciting axioms.

I doubt you can do either.

 

[phiwum]

Sorry, no. The idea of sets themselves is often taken as primitive, but "is a member of" is a very clearly-defined idea: the element in question belongs to the set. It is simple, but fully defined. All relationships are. They have to be, in fact, or trying to use them to describe anything is useless.

With due respect, traditional (ZF) set theory does not use "is a set"at all, since everything in the domain of discourse is a set. "Is a member " is undefined, with its meaning given implicitly by axioms.

In set theory (but not in PA), successor is explicitly defined, contrary to what Doronshadmi says.

ETA: this might be what you meant, but it wasn't clear to me.

 

[Nonpareil]

this might be what you meant

It was. I tried to put it in rather simpler language because I have long since learned that introducing unnecessary detail to a discussion with doronshadmi just results in a discussion that could be appropriately likened to a kudzu sprawl, so details were necessarily lost, but it was.

 

[phiwum]

It was. I tried to put it in rather simpler language because I have long since learned that introducing unnecessary detail to a discussion with doronshadmi just results in a discussion that could be appropriately likened to a kudzu sprawl, so details were necessarily lost, but it was.

Okay, my mistake. Carry on and I'll sit quietly in the corner.

 

[Nonpareil]

Okay, my mistake. Carry on and I'll sit quietly in the corner.

No, by all means, join in. It's been jsfisher locked in this cage basically on his own for the past several years. I'm just here as a vaguely-interested passerby, really.

 

[jsfisher]

Let's recap this latest thread arc.

After an extended absence, Doron returns to declare aleph₀ is simply a "number" larger than any natural number.

Doron then introduces a substitute for aleph₀, a "number" also larger than any natural number. He throws in some notational gibberish claimed to be self-evident.

Doron's introduction behaves differently from aleph₀ so therefore aleph₀ is wrong, and Doron has found a new insight into infinite sets.

The reason we've been fooled into accepting aleph₀ rather than Doron's "number" is our failure to recognize infinite sets are incomplete.

Incompleteness does not appear as a concept anywhere in the set theory axioms, in particular the Axiom of Infinity. Doron is accused of rejecting the Axiom of Infinity.

Doron denies he rejects the Axiom of Infinity. He just has a different interpretation of it. He adds words without meaning to the axiom to prove his point.

Pressed with the fact the Axiom of Infinity is expressed in predicate logic, not English, Doron again affirms he does not reject the axiom; it just needs a little adjustment.

After a round or two with predicate logic gibberish, Doron finally arrives at his tweaked version of the axiom. It now includes something, something about "successor", and Doron declares, "Therefore incomplete!"


We are now waiting for some meaning for "successor" and its linkage to completeness. It will probably be a long wait, but it would be pointless to press forward with any Doronisms while this crevasse in meaning remains.

 

[phiwum]

I'm afraid I've previously served my time in exceedingly long, pointless discussions with persons with colorful notions about set theory, diagonalization and infinity.

Obviously, I haven't given up long, pointless discussions with persons with colorful notions entirely, as current threads on this site show, but I think I'll give the math thing a pass largely.

The one question I have for Doronshadmi before I go.

1,000,000... is a place-value representation, where each digit has some finite index, yes? (I presume the "1" is in position 1, and the next position is 2 and so on, although traditionally the numbering begins at the other end, which evidently doesn't exist here.)

1 + 1,000,000,... is different than 1,000,000... .

If two numbers are different, then their place-value representation differs.

At which index i is the i'th digit of 1,000,000,... different than the i'th digit of 1 + 1,000,000,...?

 

[Nonpareil]

1,000,000... is a place-value representation, where each digit has some finite index, yes?

No. Doron has spent many days harping on the fact that the position index of the "1" in his hypothetical pseudonumber is infinite. He is defining his "place values" by places left of the redix [sic] point.

He also rejects the notion of countable infinity outright, so the entire thing is just an incoherent mess.

 

[phiwum]

No. Doron has spent many days harping on the fact that the position index of the "1" in his hypothetical pseudonumber is infinite. He is defining his "place values" by places left of the redix [sic] point.

He also rejects the notion of countable infinity outright, so the entire thing is just an incoherent mess.

I thought he said somewhere that there are countably many digits, but each digit is in a finite position from the left (which, of course, is more or less what is suggested by the notation).

In any case, however many digits, one would think that the two numbers 1,000,.... and 1 + 1,000,.... differ at some index and so one may still ask: which index?

 

[doronshadmi]

It is nice that you are talking to yourselves under your currently accepted (standard) notions about sets.

In order to be opened to non-standard notions about sets post http://www.internationalskeptics.com/forums/showpost.php?p=11264035&postcount=1249 is as an effective start.

So I am still waiting to your detailed reply about http://www.internationalskeptics.com/forums/showpost.php?p=11264035&postcount=1249.

 

[doronshadmi]

"is a member of" is a very clearly-defined idea: the element in question belongs to the set. It is simple, but fully defined.

"is a succesor of" is a very clearly-defined idea: the element in question is next to the set. It is simple, but fully defined, where http://www.internationalskeptics.com/forums/showpost.php?p=11264035&postcount=1249 shows this notion.

Once again, http://www.internationalskeptics.com/forums/showpost.php?p=11263283&postcount=1235 is wff and you (still) have no argument about it.

 

[doronshadmi]

I thought he said somewhere that there are countably many digits, but each digit is in a finite position from the left (which, of course, is more or less what is suggested by the notation).

In any case, however many digits, one would think that the two numbers 1,000,.... and 1 + 1,000,.... differ at some index and so one may still ask: which index?

Please look at http://www.internationalskeptics.com/forums/showpost.php?p=11257134&postcount=1164, http://www.internationalskeptics.com/forums/showpost.php?p=11259142&postcount=1179 and http://www.internationalskeptics.com/forums/showpost.php?p=11259463&postcount=1185.

 

[jsfisher]

"is a succesor of" is a very clearly-defined idea: the element in question is next to the set.

Great. Then you should have no trouble expressing that meaning in the form of predicate logic (either explicitly, or implicitly via additional set theory axioms).

 

[doronshadmi]

Great. Then you should have no trouble expressing that meaning in the form of predicate logic (either explicitly, or implicitly via additional set theory axioms).

By predicate logic there is Rab.

If R is interpreted as "is a successor of" (notated by "$") and a or b are sets, then {y}$y is the explicit form (for example, in the case of the set of singleton sets) and {{y}$y} is the implicit form (in the case of inductive sets).

is an axiom of the implicit form.

It has to be stressed that the notion to establish a set theory that is strong enough to deal with arithmetic and also proves its own consistency, does not hold after Gödel's work (completeness and consistency prevent each other in such axiomatic theories).

ZF(C) is an axiomatic set theory that is strong enough to deal with arithmetic.

Moreover, it silently uses {y} as a part of the axiom of infinity.

If this silence is sounded (as I do by using {{y}$y} as the implicit form of such axiom) no inductive set is complete, and since the set of natural numbers provides the ability to deal with arithmetic to ZF(C), such an axiomatic system is consistent only if inductive sets are incomplete.

 

[jsfisher]

By predicate logic there is Rab.

If R is interpreted as "is a successor of" (notated by "$") and a or b are sets, then {y}$y is the explicit form (for example, in the case of the set of singleton sets) and {{y}$y} is the implicit form (in the case of inductive sets).

Well, ok. It sounds like what you are fumbling to express is this:

where S_d(x, y) is the proposition, y is the successor of x, in doronetics. More simply expressed as a function, S_d(x) = {x}.

That, by the way, would mean your amended axiom has a tautology inserted into it. It serves no purpose, and can be removed without change to the axiom.

Moreover, you still have provided no connection between your non-standard successor function and completeness.

So far, all you have done is misused a standard term to name something that really didn't need a name.

 

[doronshadmi]

Well, ok. It sounds like what you are fumbling to express is this:

[qimg]http://www.sciweavers.org/tex2img.php?eq=%5Cforall%20x%2C%20%5Cforall%20y%20%5C%2C%20%28%20S_d%28x%2Cy%29%20%20%5CLeftrightarrow%20%28%20%28%5Cexists%20z%20%5Cin%20y%29%20%5Cwedge%20%28%20%5Cforall%20z%20%5Cin%20y%20%5C%2C%20%28%20z%20%3D%20x%29%20%29%20%29&bc=White&fc=Black&im=jpg&fs=12&ff=arev&edit=0[/qimg]

where S_d(x, y) is the proposition, y is the successor of x, in doronetics. More simply expressed as a function, S_d(x) = {x}.

What is written in http://www.internationalskeptics.com/forums/showpost.php?p=11264035&postcount=1249 has nothing to do with you wrote above.

In other words, you wrongly interpret http://www.internationalskeptics.com/forums/showpost.php?p=11266069&postcount=1275.

In order to be clearer, by {y}$y form the successor {y} is external to y, where by {{y}$y} form the successor {y} is external to y within set x (where x is notated by the outer "{" and "}").

Therefore set y is externally incomplete, where set x is internally incomplete.

In order to understand it please think about y as the set of infinitely singletons that its completeness is externally incomplete, and if this incompleteness is defined within set x (where x is notated by the outer "{" and "}") then set x is internally incomplete (and this is exactly the case of

about inductive sets).

 

[Nonpareil]

Moreover, you still have provided no connection between your non-standard successor function and completeness.

Come to that, I'm fairly certain that he hasn't even defined "completeness".

 

[doronshadmi]

Come to that, I'm fairly certain that he hasn't even defined "completeness".

A complete set is not defined by {y}$y for any y.

You simply do not wish to learn any new notion (for example: http://www.internationalskeptics.com/forums/showpost.php?p=11265933&postcount=1272), isn't it Nonpareil ?

 

[doronshadmi]

where S_d(x, y) is the proposition, y is the successor of x, in doronetics. More simply expressed as a function, S_d(x) = {x}.

jsfisher, please use

in order to define x as the set of all singleton sets.