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[MO] Is there a tetration for infinite cardinalities? (Question in MO)
I have to much to comment about this for a single post.

It might not make me popular but here we go ...

Those who know me better or longer know that I used too have many discussions and flame wars about set theory on sci.math and a few other math forums.

Despite harmfull to my reputation, I strongly believe in what I call " mathematical truth ".

This implies that I do not agree that any self-consistant axiom is necc a good axiom.

In particular I am an opponent of ZFC , AC , CH , GCH and some Cantorian ideas.

These discussions have been going on for about 15 years online and even longer offline so I cant possible rewrite all arguments here.

The funny thing is although I strongly fight ideas of say Cantor and Conway , I find these men to be very intresting in OTHER AREAS OF MATH THEY WROTE ABOUT.
But there ideas of set theory are imho well lets say " unlogical ".

Set theory might be intresting for game theory but apart from that I see it completely failing to connect to other domains of math.

It seems to be a trend that number theorists and tetration fanatics usually oppose to set theory ideas ( fully or partially ).

Although I admit some discussions went into " philosophy mode " I still defend my viewpoint.

First : the idea of limits belongs to calculus and has no place in set theory since ordinals and cardinals are not even continuous or algebraicly closed.

However since they are ordered we can take the sandwich theorem.

This answers MphLee ' s question somewhat :

w^^w = 2^(ln(w) 2^^w)
This follows from 1 < ln(w) =< w which is essentially the sandwich theorem applied to the logarithm and then using induction.

( ln(w) 2^^w = 2^^w follows easily )

SO set theory has no big connection to calculus.

I will continue to argue that set theory has no connections to other fields , although the majority insists it does.

2^x = w has no solution in set theory.

So set theory has no big connection to algebra or regression.

Number theory and geometry.

(infinite) ordinals do not relate to infinite descent ( despite being infinite )

modular arithmetic , periodic functions or properties of integers.

sin(w) is a good example of this.

Sure we can take the interpretation f(w) = sup f(w) but that would simply give sin(w) = 1.
Which is completely useless in any number theory proof attempt.

Some will argue that f needs to be strictly increasing and ordinals are used to estimate the " size " rather than the " value ".

But that does not resolve the issues.

One of the 2 most bogus things in set theory are these :

Banach - Tarski paradox.

Now it said that its not a real paradox ... but it is.

It is PROOF BY CONTRADICTION that ZFC and vitali measure is inconsistant ( in particular AC and vitali measure ).

See also this simple page :

The second one is the so-called proof of the Goodstein theorem ( about the Goodstein sequences ).

Using weird ordinals and axioms and pretending its about integers and number theory is a shame imho.


Thats just the intro about my skepticism.
I have been called the C-word and it was suggested that I do not understand even " basic things " such as cantor's diagonal ...
but I do.

Imho I do not fit in the class of those who do not understand mathematics and I find it unnoble and honorless to try to put me there as an argument.


So how about dynamics ?

Also this is at the heart of the failure of ZFC + CH + ordinals.

So we know that w^2 = w and 2^w >w.

also w! = 2^w.

And we said before that we could interpret sin(w) as 1.

But what about a function f that is close to exp^[1/2].

Then what is f(w) ??

Despite giving 2^^w I cannot think of a meaningfull interpretation of

exp^[1/2](w) and rounding or 1-periodic theta functions do not change that.

And now we are full circle :

those who like tetration and number theory wonder about

floor ( exp^[1/2] ( w ) )

and the "evil" set theorists say tetration is nonsense as defense.

Funny because Goodstein uses (integer) tetration and that proof is celebrated.

though wrong ...

So is there a cardinality between w and 2^w ( CH ) ?

Maybe its exp^[1/2](w) ...

But what the ... is that ?

And for those who like both tetration and set theory I challenge you to define it !



the master

Messages In This Thread
RE: [MO] Is there a tetration for infinite cardinalities? (Question in MO) - by tommy1729 - 12/04/2014, 12:08 AM

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