Re: My talk about Godel to the post-grads.

Mitch wrote:
On Jul 15, 3:02 pm, John Jones <jonescard...@xxxxxxx> wrote:

Whereas,
inclusive or is employed without any distinction made between 'and' and
'or'.

What? you may not notice a distinction, but most people use them quite
distinctly. 'inclusive or' is different from 'and'; check out the
truth table, or look at the axioms of logic.

I know inclusive or is treated differently to 'and' and 'or' separately. That's because it uses both 'and' and 'or' together - it makes no distinction between them.


What in this case is an intersection? Does it
separate the domain of A and B from everything else, or does it bring
them together?

the intersection is what is common to the sets A and B, only what is
in both.
the union is what is in A or B or both.

Right. So what is in common allows us to speak of two or more things. Can we talk about what is in common in the context of only one thing? as, for example, when 'x is only in A' in the definition of a union?


Yes, but what allows us to consider the elements of only both? For it
still seems as if we do not distinguish, for example, not A from B.

Let me try a different explanation. Sets are really a way of
objectifying a property. Metaphorically, a set is like a bag that you
put things in, but that evokes too much of real word objects, where
there is at most one of any thing. And so the way you seem to use that
metaphor, an item can only be in at most one set at a time, and two
bags can't 'overlap.

A better metaphor for a set is that it is like a name of a property.
We call all those items that have the property a set (thinking of them
'together'). The set of primes is really the property of being prime.
So 2, 3, 5, 7, 11, etc have the property of being prime, means we can
refer to the set of primes (which consists of 2, 3, 5, 7, 11, etc).

Those numbers that have the property of being odd can be referred to
also as a set, namely the set of odd numbers.

Notice that a number can have many properties. In this way, we are
then led to think of a number as also belonging to many sets.

Consider the number 3. It is both odd and prime. So we say it belongs
to both the set of odd numbers and also the set of primes.

An even number is -not- odd (and vice versa) so we say the set of
evens is the complement of the set of odds.

The intersection of the set of primes and the set of evens is the set
with exactly one member, {2} (I write that in the conventional set
notation).

The union of the primes and the odds turns out to be the odds along
with the number 2.

Did the union (or intersection) of two sets exist require a
preestablished connection before the elements in it are determined? I
have no idea, and the point is it doesn't really matter. We can talk
about the union as unknowns first and then decide to pick out its
contents later. That's what's nice about names and variables. You
don't have to have the object in your hand, you can just talk about
its properties.

My original objection turned out to be quite fundamental then. My original objection was that without a prior bringing together (in some way) of A and B we could not announce AuB when only A (or B) plays an actual role in it. For if we announced AuB when only A or B play a role in its formation then we give significance to at least one redundant sign.

I hoped to use your example of primes and evens here, but it doesn't reflect that other case we were dealing with where x is only in either A or B, and not both. Have you got an example?
.

Relevant Pages

  • Re: My talk about Godel to the post-grads.
    ... your objection is universal and therefore everything falls together or ... It establishes as much of a relation as union, to which I should add, ... The set of primes is really the property of being prime. ... Those numbers that have the property of being odd can be referred to ...
    (sci.logic)
  • Re: My talk about Godel to the post-grads.
    ... To address your objection that "it makes no distinction between them." ... the union is what is in A or B or both. ... The set of primes is really the property of being prime. ... My original objection turned out to be quite fundamental then. ...
    (sci.logic)
  • Re: Formulating sentences in a possibly consistent ZF
    ... You might be interested to know that Goldbach's weak conjecture, ... every integer greater than 5 can be written as the sum of 3 primes, ... single odd ... on Dedekind's property of infinity. ...
    (sci.logic)
  • Re: Formulating sentences in a possibly consistent ZF
    ... You might be interested to know that Goldbach's weak conjecture, ... every integer greater than 5 can be written as the sum of 3 primes, ... we can't define odd numbers in the same manner! ... on Dedekind's property of infinity. ...
    (sci.logic)
  • Re: sums of divisors
    ... No odd squares are both 'neat' and 'slim' ... Call a number n 'slim' if only one ... since c does not divide ... Call the primes that divide into any ...
    (sci.math)