Re: Review of Mueckenheims book.

On Mar 13, 12:46 pm, David Marcus <DavidMar...@xxxxxxxxxxxxxx> wrote:
MoeBlee wrote:
On Mar 13, 6:57 am, Aatu Koskensilta <aatu.koskensi...@xxxxxxxxx>
wrote:
On 2007-03-13, MoeBlee wrote:
However, I have not seen a formulation of the triples approach from
primitives, as set theory defines everything from the primitives 'e'
and '=' (or even from 'e' alone in some treatments). There may be a
formulation of the triples approach from primitives; I just don't know
of one.

I don't recall if one can find such a treatment in the literature, but in
any case it's trivial to provide one.

And I tooo don't imagine there would be great problems in doing so;
but it would seem to require the adjustment of some other the other
terms alongside it.

Let's look at some examples.

Me: A well-ordered set is a pair consisting of a set and an order
relation where the order relation is a well ordering.

You: A well-ordered set is just a set. Assuming the axiom of choice,
all sets are well-ordered sets.

Me: A function is a triple consisting of a domain, codomain, and a
special type of relation.

You: A function is a special type of relation.

Me: A measure space is a triple consisting of a space, a sigma
algebra, and a measure.

You: ?

Me: The identity on Z and the inclusion of Z into R are different
functions.

You: The identity on Z and the inclusion of Z into R are the same
function.

Me: The identity on Z is a surjection. The inclusion of Z into R is
not a surjection.

You: We can't say whether a function is a surjection unless we say
surjection onto what.

Me: The identity on S^2 is not homotopic to a constant.

You: We can't say whether the identity on S^2 is homotopic to a
constant unless we say what codomain the homotopy can use.

Your imaginary dialogue doesn't interest me. Moreover, as to well
orderings, what I said about that is in the thread in which we
discussed that; and you must recognize that I made a distinction
between what the usual ways of speaking are as opposed to what I think
would be better terminology. I was, and am, very clear to say that my
simply disliking a certain informal terminology or even formal
terminology does not in the least deny that that terminology is indeed
usual or even standard or even that, in that particular case, I would
use terminology that would cause confusion due to a clash with usual
or standard terminology.

Time and interest permitting, I am happy to provide the formal (semi-
formal, in the sense of using English language handles) definitions I
use, starting from primitives.

As to the question of formalizing a triples definition from
primitives, and given some set of axioms, I have no objection to it; I
only pointed out there would have to be adjustments from the set
theoretical definitions of some other terms as well.

Originally, I said that in set theory the identity function on omega
is a function from omega into the power set of omega. That is correct.

Then you claimed that my definition of 'is a function' is personal,
and, as I recall, you said "idiosyncratic". But that is not correct,
since my definition is the standard definition in set theory, which is
even the stated context of my original remark. Also, I stated that the
definition is found in many other books on mathematics. And I quoted a
few examples for you; and you quoted one yourself. Moreover the three-
place definition (as long as it specifies a relation rather than a
'rule' or just generally a 'correspondence') is also used in set
theory and even entails the one-place version.

You first recognized that you were including one of your own quotes as
an example of the set theoretic definition, but at a point later, each
time you you gave a summary of your original survey of quotes, you
left out mentioning that you had conceded that one of the examples is
the defintion I mentioned. I called you on that fact, and you still
never recognized the discrepency in your own presentation.

You tried to minimize in general terms any examples from an analysis
book on the grounds that all functions are there are real valued
anyway. And that is a specious argument since, (1) analysis books need
not limit themselves to real valued functions, (2) even if every
function mentioned in such books were real valued, it doesn't follow
that each codomain of interest is R, since there may be proper subsets
of R that may be codomains of interest, (3) the books I mentioned did
not confine the definitions to real valued functions, and one of them
was even explicit to say that the definition of 'relation' (which was
the basis for the definition of 'function') pertains to any objects
whatsoever, not even necessarily numbers, let alone real numbers.

You claimed that Halmos is to be understood as specifying that
functions are triples, not certain kinds of relations, even though
Halmos explicitly says that functions are (certain kinds of) relations
and even explicitly states the requirements for proving something is a
function, which are requirements of showing it is a certain kind of
relation and not a triple.

You made a snide aside, not even to my (cyber virtual) face about
"semi-cranks" taking a posting break, and later snidely explained our
difference of views on what you claim to be my inability to read a
math book. Then, even when I explained to you that I do not wish to
discuss my personal background on the Internet, especially in a Usenet
thread, and especially with you (given your snideness), you still
continued to hector on the point, and still even after I allowed you
that you may consider that I do not represent that anything I've said
has any authority or influence of oral classroom teachings whatsoever.

I made one error in not recalling that Browder's is the three-place.
However, as I quoted, Browder's three-place is still the set theoretic
three-place, even as he makes explicit that it is the graph and that
that it is in keeping with set theoretic trends.

And, all of that stemming from your original remark that my definition
is idiosyncratic, which remark of yours is plainly incorrect. And you
continue with gratituous combativeness, as you ask asinine questions
such as "how many set theorists have you talked with?" in response to
a posting of mine in which you mention no specific objection. And
rather than direct yourself to a possilby constructive, substantive,
and interesting discussion about different proposals for
formalization, you continue still with an asinine confrontational
rhetorical device such as an imaginary diaglogue that puts words in my
mouth, either out of context from crucial qualifications I had
mentioned or to project what you think my answers (would be?) to
certain questions.

I wish there were some nicer way to say this that has the same direct
honesty, but you are by this time presenting like a real jerk, and
over a dispute in which you were plainly incorrect from the start and
have dug your hole deeper and deeper (Halmos, indeed).

Oh, but am I hypocritical to call you a 'jerk' when I just complained
of your own insults? No, because I would not fault you for having
insulted me had I previously insulted you. But I didn't and you did.

Now, if you wish to engage in a productive exchange, then just give a
holler. But, in the meantime, I'm not inclined to indulge such
imaginary dialogue as you just posted.

MoeBlee






.

Relevant Pages

  • Re: proof of undecidability of halting problem
    ... Within these primitives we find well-known principles from Foundations ... I give an Algebra by which theorems are stated as ... also applies to problems of Set Theory e.g. construction of a Quine ... Paradoxes: I formally represent "This is false." ...
    (sci.logic)
  • Re: An uncountable countable set
    ... That's pretty much what we do in set theory. ... number of primitives can only address a finite number of properties. ... The set of definining formulas of Z set theory is countably infinite. ... You say it contradicts set theory, but I imagine that depends on how you ...
    (sci.math)
  • Re: Set theory and identity theory
    ... set theory, ... Didn't you like my considerations? ... there's a REASON why considering "identity" a logical primitive (in FOPL ... a finite number of primitives provides some kind of special case. ...
    (sci.logic)
  • Re: Kuratowski Ordered Pair
    ... the other concept Hamilton couples ... One can take set theory as the basics and then define ordered ... pairs, and n-tuples, in the Kuratowski manner. ... a digital computer as granted and conceive n-tuples there as primitives. ...
    (sci.math)
  • Re: Kuratowski Ordered Pair
    ... the other concept Hamilton couples ... pairs, and n-tuples, in the Kuratowski manner. ... a digital computer as granted and conceive n-tuples there as primitives. ... However, if one is to claim to have a better system than set theory, ...
    (sci.math)
Loading