Re: Set Theory: Should You Believe

If 'l.i.m.i.t' is the sign which is employed when certain manoevures
need to be made in mathematics, then we need not associate any meaning
with the letter sequence, or word: if the word 'limit' has any meaning
it would be a meaning we associate with familiar objects and
behaviours. But if mathematics does not deal with familiar objects and
behaviours then its term 'limit' cannot convey meaning. As mathematics
does deal with familiar objects and behaviours then we can safely
assume, if mathematics is consistent in its use of terms, that we can,
as I said before, understand the term in its simple, originary meaning.
Even if the symbols with which the term 'limit' is associated are
different, the intuitive, metaphysical, familiar behaviours or acts
which the term invokes would be similar. That is, the term limit must
always refer to simple behaviours and objects to have meaning.

By saying 'the number which is not a limit of a sequence of numbers is
not found "in a sequence of numbers"', I do not mean to say that
numbers are transferable between sequences. Numerals are transferable
because they are not bound to any application.

Its difficult to imagine how a sequence of numbers can be a sequence of
'numbers'. It seems that a different application is made for each
'number', and that the results are gathered and placed in an order that
is largely visual and impressionable, rather than 'mathematical'. In
which case, not least because we are trying to transfer numbers between
applications, it seems that we have numerals, and not numbers. And a
sequence is pictorial, which is why I said it might be a representative
structure, and strictly not a mathematical device. Regarding your point
about grasping a number, of course I think there is some confusion
where numbers and numerals have the same form. Also, as you say 'there
is no number to be grasped' unless we make one.


Perhaps a different notation will be helpful. Suppose we represent
the sequence of numbers (or numerals) as a_1, a_2, a_3, etc.
We can now specify the sequence by specifying what a_n is to
refer to for each number (or numeral) n, e.g., in this particular
case we can say that a_n = 1 for each n. In the other example
(below) that for each n, a_n = n. This would fix the position of each
element, because each element has a unique nametag, which not only
names the element but gives its place in the sequence.

Yes, if each element has a different name tag then I can distinguish
the start of the sequence from the end; or if not exactly the 'end',
then the identifying act which identifies the 'limit' of the sequence.
Just off the cuff, I think the problem with this idea is that I need to
place the tags in order, if I want to identify the object to which they
are attached. We could just have sequence of numbers as they arise in
addition, 1,2,3,4 seems to identify element and position... but does it
identify an element with its position? After all, the tags are
arbitrary.

There may be a problem here in distinguishing between "the sequence"
as an object concretely given, i.e., numerals, and "the sequence"
as the (presumably existing) object referred to.


Yes, it seems that sequence is more like an act that yields a pictorial
form or order. In which case the act that constructs this order is not
the act of making the numbers that are said to compose it. This might
mean that I have no means of mapping 'number' to 'position'.


Do we agree that the limit of this sequence is a member of this sequence?

Not if my examination above, top, was correct.

I suggest that the new notation meets the objection of indeterminacy of
position.

Yes, this needs some more exploration. See also my response above.

.

Relevant Pages

  • Re: Choice sequences, intuition, etc
    ... specifying a different method of constructing the sequence, ... Regarding Bishop, ... axiom systems for mathematics that are weaker. ... Somehow the distinction in meaning between an implication ...
    (sci.logic)
  • Re: Set Theory: Should You Believe
    ... need to be made in mathematics, then we need not associate any meaning ... By saying 'the number which is not a limit of a sequence of numbers is ... applications, it seems that we have numerals, and not numbers. ...
    (sci.logic)
  • Re: X = Z
    ... travel a path and return to its beginning, a goal of mathematics is ... that have otherwise generally canonical meaning. ... the fundamental discovery of truth in mathematics, ... appear on the subject of 'primes' above and below mine, ...
    (sci.math)
  • Re: X = Z
    ... travel a path and return to its beginning, a goal of mathematics is ... that have otherwise generally canonical meaning. ... Fashion has nothing to do with truth. ... appear on the subject of 'primes' above and below mine, ...
    (sci.math)
  • Re: an true information theory
    ... If a message has meaning then it has structure and will show a pattern. ... if there is a pattern in the sequence of numbers. ... solve the halting problem, which is uncomputable. ...
    (sci.math)