Re: Logarithm of transfinite numbers


Virgil wrote:
In article <1142038371.083985.265160@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
matt271829-news@xxxxxxxxxxx wrote:

Virgil wrote:
In article <MPG.1e7b97e77147e4b498aab7@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

matt271829-news@xxxxxxxxxxx said:
What value, if any, can be ascribed to the logarithm of aleph_n?

Take log_2(aleph_0) as an example.

log_2(aleph_0) can't be finite, and it can't be bigger than aleph_0, so
it has to equal aleph_0.

But log_2(aleph_0) = aleph_0 implies 2^aleph_0 = aleph_0, whereas in
fact 2^aleph_0 = aleph_1.

Where did it go wrong?



You paid attention to Cantorian set theory hocus pocus, to begin with. ;)

You make a good point. Let's see what others had to say.......

Logarithms have definitions. For positive real numbers a
the natural logarithm of x is defined to be the area under f(x) = 1/x
between x = 1 and x = a,

ln(a) = Int_1^a 1/x dx.

Then the logarithm of a to base b, where a and b is a positive reals and
b is not 1 is defined by

log_b(a) = ln(a)/ln(b)

Note that the natural log of a number is only defined for finite
positive reals, so that other logs are equally limited.

There is no a priori reason why logs of things which are not real
numbers need make any sense at all, and trying to make sense of logs of
"infinite numbers" is silly.

No more silly than trying to make sense of the logarithms of negative
numbers, which is not at all silly. No more silly than trying to make
sense of other operations on transfinite numbers - such as addition,
multiplication and exponentiation.

Or do you just think the whole idea of tranfinite numbers is "silly"?

Transfinite cardinals and ordinals are far from silly, but the axioms
allowing them and their definitions and properties are carefully worked
out, checked and double checked to be sure that they do not incorporate
any self-contradictions.

If an when TO presents a carefully worked out system whose properties
are equally deducable from axioms and definitions and are equally free
of self-contradiction, only then will TO have anything at all
mathematical.

You said it was silly even to *try*. That is what I disagree with!

.

Relevant Pages

  • Re: Logarithm of transfinite numbers
    ... positive reals, so that other logs are equally limited. ... There is no a priori reason why logs of things which are not real ... No more silly than trying to make sense of the logarithms of negative ... any self-contradictions. ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... the natural logarithm of x is defined to be the area under f= 1/x ... positive reals, so that other logs are equally limited. ... There is no a priori reason why logs of things which are not real ... No more silly than trying to make sense of the logarithms of negative ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... Then the logarithm of a to base b, where a and b is a positive reals ... There is no a priori reason why logs of things which are not real ... No more silly than trying to make sense of the logarithms of negative ... Virgil and I have an ongoing argument. ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... positive reals, so that other logs are equally limited. ... There is no a priori reason why logs of things which are not real ... No more silly than trying to make sense of the logarithms of negative ... Virgil and I have an ongoing argument. ...
    (sci.math)
  • Re: Fings we was lernt rong in skool (Was Basrawis n all that cop)
    ... courses at university) still study logarithms, ... Natural logs are the answer to many equations in those fields. ... They are an intrinsic part of higher mathematics. ... Logs base ten are pretty useless nowadays, though, save for graph ...
    (alt.usage.english)