Re: Is the empty set a number?

On 10 Apr, 04:48, jonas.thornv...@xxxxxxxxxxx wrote:
On 9 Apr, 00:05, lwal...@xxxxxxxxx wrote:





On Apr 7, 2:26 pm, jonas.thornv...@xxxxxxxxxxx wrote:.

Can you tell me if two empty sets equals one empty set i would be
greatful for an answer.

By the Axiom of Extensionality in ZFC, all empty sets are equal.

It appears that the OP has one of two concerns:

1. Is zero really a number?
2. Is the empty set really a set?

And these two questions have distinct answers.

1. Many of the others have already discussed to which objects
we assign the property of numberhood in this thread. But the
bottom line is that any ring must have an additive identity,
and that element is zero. Indeed, every semigroup can be made
into a monoid by simply attaching such a neutral element.

To standard mathematicians, the unqualified word "number"
refers to an element of C, the set of complex numbers. Thus
zero is a number since 0eC.

One might ask what the set of all numbers is to the OP. If
we let J (for Jonas) be the set of all numbers that the OP
accepts, then clearly J is a proper subset of C, since 0eC
but yet ~0eJ.

So what numbers are in J? I doubt that the OP accepts the
existence of negative numbers, since then one would have
to wonder what -1+1 is.

Well that would be dent to no accept negative numbers of course i do.
What i say is that the result from the transaction 1-1 is not a
number, and that is quite another issue. If you accept this and build
your numbersystem and architecture supporting this. You do not have to
think about division by zero. And it still will give perfectly valid
result for any calculation.

Historically the Greeks and Romans,
just like the OP, denied the existence of zero, but the
Sumerians, Mayans, and Hindus all had symbols for zero.

Yes i am aware of that, and i think it ís a problem inherited by our
logic to want to put number to nothing. Because nothing really do not
have a *value*.

Representing a number as base dependent postional value sets works
very good for any base.

Here is some expamples in "DECIMAL" base using positional value
representation and as you can guess 0 is missing representation in
this system.

-2000500009=-{[A,2][5,5][1,9]}
9000020005={[A,9][5,2][1.6]}
0,50000000009=ooops again{[-1,5][-B,9]}should of course be A1,9
-1.90005=-{[1,1][-1,9][-5,5]}





The largest set one can have without the existence of zero
is the set of unsigned reals R+, which is labeled by a
script P in Metamath. Notice how Metamath develops the
unsigned fractions and unsigned reals via Dedekind cuts
well before developing zero and signed numbers. This
matches the historical development, where Pythagoras
discovered sqrt(2) over 2000 years before Cardano
introduced negative numbers.

Kronecker said that "God created the integers" -- but it's
uncertain whether he meant the positive natural numbers or
the signed integers. But of course, one trick is simply
to let a signed integer simply be an equivalence class of
ordered pairs of natural numbers, as is usually done, so
that 0 = {(1,1),(2,2),(3,3),...}.

2. But if you deny the existence of the empty set, then
you have a deeper problem than if you merely deny the
numberhood of zero. For the Axiom of Foundation (AKA
Regularity) states that every set is based on the empty
set, in that every set has the empty set as an element of
its transitive closure. So if you don't want an empty set
then you must deny Foundation/Regularity.

One sci.logic poster, Zuhair, also wanted to come up with
a set theory once in which there is no empty set. But
unfortunately, he was not able to come up with such a
theory in a way that it would be consistent.

You try this- Dölj citerad text -

- Visa citerad text -- Dölj citerad text -

- Visa citerad text -

.

Relevant Pages

  • Re: Is the empty set a number?
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  • Re: Is the empty set a number?
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