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Set Theory: Should you believe?
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Virgil
science forum Guru


Joined: 24 Mar 2005
Posts: 5536

PostPosted: Fri Jul 21, 2006 3:33 am    Post subject: Re: Set Theory: Should you believe? Reply with quote

In article <rimvb29t289hak5l58g1kadn0rkev5qnvi@4ax.com>,
Lester Zick <DontBother@nowhere.net> wrote:

Quote:
On Thu, 20 Jul 2006 12:56:26 -0600, Virgil <virgil@comcast.net> wrote:

In article <n68vb2p91g8m3nbbp0k4v7qbit7m6jfutn@4ax.com>,
Lester Zick <DontBother@nowhere.net> wrote:

On Wed, 19 Jul 2006 19:47:02 -0600, Virgil <virgil@comcast.net> wrote:

In article <tvbtb29qjsde0k18m272crf3092esavcsu@4ax.com>,
Lester Zick <DontBother@nowhere.net> wrote:

On Wed, 19 Jul 2006 14:47:00 -0600, Virgil <virgil@comcast.net> wrote:

You have not read my reference to " logical tautologies" correctly.

If, for example. "P and not P" would qualify as "false" does your
gobledegook require its negation, "P or not P", to be true?

"P and not P" is only universally false because it provides no
mechanical basis for alternatives since any "not (P and not P)"
converts into itself "not P and P".

Not in any respectable logic it doesn't. According to de Morgan's laws,
"not (P and not P)" is logically equivalent to "P or not P".

It may be equivalent to lots of things. The issue is whether it
converts into itself mechanically.

According to de Morgan, and others,
"not (P and not P)" and
"P or not P"
convert quite mechanically into each other but
"not (P and not P)" and
"not P and P"
do not convert into each other in any way at all.

I get dizzy just trying to read all this

Then you have no business trying to deal with formal logic, as this is
quite simple compared to most of it.
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Virgil
science forum Guru


Joined: 24 Mar 2005
Posts: 5536

PostPosted: Fri Jul 21, 2006 3:37 am    Post subject: Re: Set Theory: Should you believe? Reply with quote

In article <jh00c2tp40c9o621j2u2cim6vrgs163cuh@4ax.com>,
Lester Zick <DontBother@nowhere.net> wrote:

Quote:
On Thu, 20 Jul 2006 13:12:21 -0600, Virgil <virgil@comcast.net> wrote:

In mathematics, all assumptions (axiom systems) are merely conditional,
to see what will follow from them. When what follows proves useful or
interesting, one tends to codify those assumptions. but that never
requires that one claims them true is any absolute sense. Such
assumptions are always "what if's".

It's clear in faith based math

"Faith based"? There is no "faith" required for axiomatic based
mathematics, only logic.
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Virgil
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Joined: 24 Mar 2005
Posts: 5536

PostPosted: Fri Jul 21, 2006 3:42 am    Post subject: Re: Set Theory: Should you believe? Reply with quote

In article <q810c29ku2cuf8h8u0p22semlvsgg19v5o@4ax.com>,
Lester Zick <DontBother@nowhere.net> wrote:

Quote:
On Thu, 20 Jul 2006 13:02:37 -0600, Virgil <virgil@comcast.net> wrote:

In article <utavb2tsrf1vme8aijr4a59bp5q450ks98@4ax.com>,
Lester Zick <DontBother@nowhere.net> wrote:

On Wed, 19 Jul 2006 21:32:14 -0600, Virgil <virgil@comcast.net> wrote:

In article <48ptb2h9tn62b5qq2hifgras3vakbotcnn@4ax.com>,
Lester Zick <DontBother@nowhere.net> wrote:

A good beginning discussion of the subject of mathematical definitions
is in Suppes's 'Introduction To Logic'. But in order not to inhibit
the
metastasis of your own convictions, I recommmend that you not read
such
books.

Well if there's one thing I detest more than assumptions of truth it's
metastasis of convictions when one is actually dealing instead with
demonstrations of truth.

As Zick has not demonstrated any truths

Neither have you, sport.

AS I am the one doubting the existence of any such things as absolute
truths or absolute falsehoods,

A veritable doubting Thomas.

my lack of demonstrating the
existence of
any such thing supports my position.

Well let's just say your lack of production in this regard doesn't
support much of anything including your position.

As Zich is the one affirming their existence, his lack of demonstration
tends to weaken his position.

But you've already admitted my general claim defies your critical
capacity.

Zick again exhibits his penchant for seeing things which do not exist.
I am critical of your general claim, as you have not been able to
bolster it with anything other than mere restatement of the claim itself.
Quote:

All we've dealt with so far is set theory as
a faith based institution of doddering ineptitude.

Don't be so hard on yourself, Zick.
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David Bernier
science forum Guru Wannabe


Joined: 01 May 2005
Posts: 101

PostPosted: Fri Jul 21, 2006 3:53 am    Post subject: Re: Set Theory: Should you believe? Reply with quote

Aatu Koskensilta wrote:
Quote:
Lee Rudolph wrote:
[...]


Quote:
I agree (based on observations of number theorists) that "one does
not say" that sort of thing, and I am open to being persuaded (indeed,
I am predisposed to be persuaded) that if a school of mathematicians
got into the habit of saying that sort of thing (while continuing to
do mathematics) then we (and possibly they) might want to say that
what they were doing (though still mathematics, and possibly very
fine mathematics) was no longer "number theory" (or, weaker, no longer
*just* "number theory"): but I don't see such a response
is self-evidently right.

That certainly would be the natural reaction. In fact, there already is
a discipline of mathematics where one can expect to hear such things. No
one calls it "number theory". It seems highly unlikely that there ever
could be a school of mathematics in which studying e.g. structures in
which the arithmetical consequences of ZFC hold was called "number
theory". That sort of a terminological shift would require a major
upheaval in the way people think about natural numbers.

In other words (I guess), is there anything more than historical
chance and prejudice behind the feeling (which I certainly share,
but don't feel particularly justified in sharing) that "natural
numbers should be *categorical*, dammit"?

It's a basic property of our conception of the natural numbers that they
don't bifurcate into a multitude of non-isomorphic structures. There are
some people of ultra-intuitionist and ultra-finitist peruasion -
Esenin-Volpin and Edward Nelson come to mind - who do think that there
are many different natural number lines and reject the ordinary
conception of the natural numbers as incoherent or unjustified. This
just goes to show, once again, that it is to a large extent a matter of
personal preference and inclination what one finds convincing and
coherent, and that even in mathematics there probably is no principle
someone competent hasn't doubted or rejected.

I consulted the web site ``Earliest Known Uses of Some of the
Words of Mathematics", for which I can't find the author.

The listings for terms beginning with the letter C can be found here:
http://members.aol.com/jeff570/c.html

A quote from the entry for CATEGORICAL (axiom system) follows:

``Thus in his The Loss of Certainty (1980, p. 271) Morris Kline wrote:

Older texts did "prove" that the basic systems were categorical;
(...) But the "proofs" were loose (...) No set
of axioms is categorical, despite "proofs" by Hilbert and others.

This remark was corrected by C. Smorynski in an acrimonious review:

The fact is, there are two distinct notions of axiomatics and,
with respect to one, the older texts did prove categoricity and not
merely "prove".

[This entry was contributed by Carlos CÚsar de Ara˙jo.] "

One suspicion I have is that Smorynski's comment is related
to second order logic as distinguished from first order logic.

If someone could elaborate on the meaning of what Smorynski wrote,
I'd appreciate it.

David Bernier
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Nam Nguyen
science forum addict


Joined: 15 May 2005
Posts: 50

PostPosted: Fri Jul 21, 2006 6:05 am    Post subject: Re: Set Theory: Should you believe? Reply with quote

Rupert wrote:
Quote:

As I said, Nam Nguyen seems to be using "absolute truth" to mean
"something that is true independently of which semantics we use".

Imho, formally, it's near impossible to *directly* define "absolute
truth". On the other hand, if we *indirectly* define it as: anything
that is relative is not absolute, then we have a good chance to
indirectly understand "absolute truth". And relativity is something
we could easily define up-and-down the ladder of mathematical
introspection. For instance, the truth of 1 + 1 = 0 is relative
to what formal system we choose, hence it can't be an absolute truth.
Or the truth of (x=x) would depend on what logical system that's being
assumed, hence it's not an absolute truth. etc...

Imho, mathematical relativity could be grouped into 2 groups:

a) interpretation/semantic-based relativity: this is when a truth value
would depend on a reasoning being's (model) interpretation, or
(semantic) interpretation.

b) knowledge-based relativity: this is when knownability of the truth
value would depend on the knowledge on the reasoning being. For
instance, without loss of generality, let's k be a number so big
that Prime(k) is unknown to human being, then Prime(k) is knowledge-
relative. (If the theory is PA, then it's possible that PA can be
inconsistent and in which case Prime(k), for there would be no
model. But PA's shortest inconsistency-proof might be even
longer than proof of Prime(k). Hence "Prime(k) is true" is a
knowledge-based relative truth.)

And again, the truth that is relative is not an absolute truth.

Quote:
Obviously no such thing exists.

I wouldn't go that far though.


--
-----------------------------------------------------

What we call 'I' is just a swinging door which moves
when we inhale and exhale.
Shunryu Suzuki
----------------------------------------------------
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Rupert
science forum Guru


Joined: 18 May 2005
Posts: 372

PostPosted: Fri Jul 21, 2006 9:38 am    Post subject: Re: Set Theory: Should you believe? Reply with quote

Nam Nguyen wrote:
Quote:
Rupert wrote:

As I said, Nam Nguyen seems to be using "absolute truth" to mean
"something that is true independently of which semantics we use".

Imho, formally, it's near impossible to *directly* define "absolute
truth". On the other hand, if we *indirectly* define it as: anything
that is relative is not absolute, then we have a good chance to
indirectly understand "absolute truth". And relativity is something
we could easily define up-and-down the ladder of mathematical
introspection. For instance, the truth of 1 + 1 = 0 is relative
to what formal system we choose, hence it can't be an absolute truth.
Or the truth of (x=x) would depend on what logical system that's being
assumed, hence it's not an absolute truth. etc...

Imho, mathematical relativity could be grouped into 2 groups:

a) interpretation/semantic-based relativity: this is when a truth value
would depend on a reasoning being's (model) interpretation, or
(semantic) interpretation.

b) knowledge-based relativity: this is when knownability of the truth
value would depend on the knowledge on the reasoning being. For
instance, without loss of generality, let's k be a number so big
that Prime(k) is unknown to human being, then Prime(k) is knowledge-
relative. (If the theory is PA, then it's possible that PA can be
inconsistent and in which case Prime(k), for there would be no
model. But PA's shortest inconsistency-proof might be even
longer than proof of Prime(k). Hence "Prime(k) is true" is a
knowledge-based relative truth.)

And again, the truth that is relative is not an absolute truth.


All right, so this is one way of understanding what "absolute truth"
means. On this interpretation, in my view the question about whether
absolute truth exists is trivial. It is obvious and uninteresting that
no sentence can be true regardless of which semantics we use.

Quote:
Obviously no such thing exists.

I wouldn't go that far though.


Why not?

Quote:

--
-----------------------------------------------------

What we call 'I' is just a swinging door which moves
when we inhale and exhale.
Shunryu Suzuki
----------------------------------------------------
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Aatu Koskensilta
science forum Guru Wannabe


Joined: 17 May 2005
Posts: 277

PostPosted: Fri Jul 21, 2006 11:53 am    Post subject: Re: Set Theory: Should you believe? Reply with quote

David Bernier wrote:
Quote:
The listings for terms beginning with the letter C can be found here:
http://members.aol.com/jeff570/c.html

A quote from the entry for CATEGORICAL (axiom system) follows:

``Thus in his The Loss of Certainty (1980, p. 271) Morris Kline wrote:

Older texts did "prove" that the basic systems were categorical;
(...) But the "proofs" were loose (...) No set
of axioms is categorical, despite "proofs" by Hilbert and others.

This remark was corrected by C. Smorynski in an acrimonious review:

The fact is, there are two distinct notions of axiomatics and,
with respect to one, the older texts did prove categoricity and not
merely "prove".

[This entry was contributed by Carlos CÚsar de Ara˙jo.] "

One suspicion I have is that Smorynski's comment is related
to second order logic as distinguished from first order logic.

Yes. The categoricity proofs by Hilbert, Dedekind and others would today
be said to establish the categoricity of certain second order theories.
There are also categorical first order theories.

--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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