Re: Cantor's definition of set

On Oct 26, 9:37?pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Oct 26, 1:29 pm, John Jones <jonescard...@xxxxxxx> wrote:

On Oct 26, 2:59?am, G. Frege <nomail@invalid> wrote:
1,2,3,4,5... is often portrayed as numbers. But aren't they examples
of the signs we use to portray numbers, and aren't these signs simply
arranged in a sequence and not numbers after all? For I cannot use
1,2,3,4 ... mathematically. 1,2,3,4 ... does not occur in any
mathematical calculation. I can find 1, and 2 and 3 and 4 in a
calculation, but in finding them don't I merely find the signs and not
the numbers themselves? What I am saying is, you can't pull a number
out of the application that generates it. It would seem, if this is
true, that a set of numbers is an impossibility.

'2' is a numeral. 2 is a number.

I don't see your difficulty with this.

The second major problem which simply won't go away, is this: A set is
the concept of a particular collection or group.

Maybe you should distinguish between whether it's a concept of a
collection or whether it IS the collection.

At least I have seen
a set described as either a collection or a group. Now a collection
does not support sequence: in fact if a collection could be a
sequence, it would be a sequence and not a collection.

No, that just doesn't follow. A sequence is a certain KIND of set.
There's no conflict in a set being of a certain KIND.

Don't get me
wrong here. I can have a set of sequences, but the the set itself, on
its own merits, cannot support a sequence. I must establish the
presence of a sequence independently of its membership in a set.

I dont' see any reason we MUST do that. We prove that f is a sequence
by proving that it has the property mentioned in the definition of
'sequence', and that definition ultimately reverts to the sole
primitive 'e'.

This,
I advance, is another reason why I cannot have a set of numbers.
Numbers in a collection are like numerals in a sack, like lottery
'numbers'

I don't envy you the knots you've tied yourself in.

MoeBlee


---
'2' is a numeral. 2 is a number.

The number 2 is always generated by an application, so I cannot
propose simply 2. I cannot say 2 is a number and leave it at that. I
must specify, and not simply assume, an application that generates it.

Even if numbers existed in a third Platonic realm as stand-alone real
entities, I would not be able to recognise them as numbers except by
employing an application, like counting. But in that case, I make
numbers myself. So there is no Platonic realm of numbers.

A sequence is a certain KIND of set.
There's no conflict in a set being of a certain KIND.

A sequence and a collection are mutually exclusive I would have
thought - a collection is indifferent to order. 'A collection of coins
of increasing denomination' does not establish a relationship between
'collection' and a sequenced 'increase'. The collection of coins would
be quite indifferent as to coin order.

.

Relevant Pages

  • Re: Cantors definition of set
    ... I would not like to say that they have anything in common. ... of the signs we use to portray numbers, ... arranged in a sequence and not numbers after all? ...
    (sci.logic)
  • Re: Cantors definition of set
    ... of the signs we use to portray numbers, ... arranged in a sequence and not numbers after all? ... There's no conflict in a set being of a certain KIND. ... The collection of coins would ...
    (sci.logic)
  • Re: Cantors definition of set
    ... of the signs we use to portray numbers, ... arranged in a sequence and not numbers after all? ... There's no conflict in a set being of a certain KIND. ... The collection of coins would ...
    (sci.logic)
  • Re: OPPOSITE OF all coin sequences are computable to infinite length ?
    ... "There is no maximum to the number of coins in any given oo coin ... Let be an enumeration of all Turing machines (equivalently, ... Turing machine started with the k'th input tape state. ... with the sequence defined above as a counter ...
    (comp.theory)
  • Re: OPPOSITE OF all coin sequences are computable to infinite length ?
    ... "There is no maximum to the number of coins in any given oo coin ... Let be an enumeration of all Turing machines (equivalently, ... Turing machine started with the k'th input tape state. ... with the sequence defined above as a counter ...
    (sci.math)