[MO] Is there a tetration for infinite cardinalities? (Question in MO)
Hi, perhaps this is of interest for someone


Gottfried Helms, Kassel
I found some related questions that came up in these days from the user Ali Sadegh Daghighi. It is about the combinatorial meaning of tetration...

[MSE]What combinatorial quantity the tetration of two natural numbers represents?

Another interesting related question is the following.

[MSE]Are there non-equivalent cardinal arithmetics?

At the end here there is another sequence of operations over the cardinals: seems the low-Hyperoperations sequence.

[MSE]How large is an uncountable regular cardinal which is closed under arbitrary fast operators?

MSE MphLee
Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)
S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)
2^^w = the first inaccessible ordinal.

This is because 2^x = x is not accessible and 2^^w is the smallest such x.

w^^w = 2^(ln(w) 2^^w) = 2^^w = the first inaccessible ordinal.

2^^(2^^w) = the second inaccessible ordinal.

2^^(2^^w) = 2^^(w^^w) = w^^(w^^w).


w^^^w = the w th inaccessible ordinal.




the master
(12/01/2014, 11:55 PM)tommy1729 Wrote: 2^^w = the first inaccessible ordinal.

This is because 2^x = x is not accessible and 2^^w is the smallest such x.

w^^w = 2^(ln(w) 2^^w) = 2^^w = the first inaccessible ordinal.

2^^(2^^w) = the second inaccessible ordinal.

2^^(2^^w) = 2^^(w^^w) = w^^(w^^w).


w^^^w = the w th inaccessible ordinal.

- Sheldon
I'm not really sure that is that easy...
The problem is that cardinals and ordinals are different and we need a good recursive definition of the tetration over the cardinals.

I mean, is true that \( \aleph_0 =\omega \) but we have also that every transfinite ordinal below \( \omega_1 \) is countable and thus has cardinality \( \aleph_0 \)

\( |\omega+1|=\aleph_0 \)

and I guess that we always have

\( |\omega\uparrow{}^{\alpha}\omega|=\aleph_0 \)

For example, all the epsilon ordinals are countable.

Back to the topic. Tommy, when you say inaccessible ordinal you mean "strongly inaccesible" cardinals right? The ones that give us models of ZFC?

I don't even understand how you can take the natural logarithm of a cardinal number...

Note also that cardinal and ordinal exponentiation aren't the same.

MSE MphLee
Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)
S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)
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 : https://sites.google.com/site/tommy1729/...urable-set

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
I respect the philosophical points of view... but i really think that this drag us off topic... is not the place to discuss about this...

Is like if I start a new thread where I ask how old are the users of the forum and you start to argue about the existence of the time itself and conclude sayng that time doesnt exist thus everyone is 0 years old (QED)...

Really, I appreciate the philosophical threads but... but...

We are talking about Cardinal arithmetic, that is defined inside a set theory (ZFC)...
you can't come here with a mysterious proof that use other deduction rules and that uses an implicit rejection of the set theory framework.

Anyways I should note that this sentence is wrong...

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

Doner and Tarski completely extended the Hyperoperations in to the transfinite ordinals!

That's because Hos are defined via recursion, so they just used the transfinite recursion over the ordinals.

At the end.. I recognize the existences of the constructivism and the finitism near the classical mathematical frameworks such as ZFC, NBG, MK ecc. ecc.

The russel paradox and the fall of the Cantor's theory originated a wide range of solutions including some that refuse the existences of cardinalities bigger than \( 2^{\aleph_0} \) (like constructivism) and some that totally refuse actual infinity (like intuitionism for example).

Said that...I still feel this wide range of differences as ideological and philosophical differences.
That's why I guess that we are going off topic and that the original thread was started to talk about the Cardinals defined inside a theory that admits different cardinality as ZFC.

MSE MphLee
Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)
S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)
Doner and Tarski can not define exp^[1/2](w) and its cardinality , right ?

Do Doner and Tarski give a different answer to 2^^w ?

I still challenge to give exp^[1/2](w) even under the assumption that ZFC is good.

Sorry for going a bit offtopic.
Guess I have emotional Luggage concerning this.

Im not 100% constructivist for clarity.


(12/04/2014, 01:20 PM)tommy1729 Wrote: Doner and Tarski can not define exp^[1/2](w) and its cardinality , right ?
The problem is not if they are able.. probably they weren't but as far as I know the didn't even tried, at least not in their work about the hyperoperations over transfinite ordinals.

The reason is clear, it was about transfinite ordinals not cardinals or fractional iteration... so your problem about the half iterate of the ordinal exponentiation has nothing to do with their work.

When you ask something like that, it should be clear, we have to be precise and define our words.
-What exp means?
-What half iterate means?
-what \( \omega \) means?

The answers to this questions involve a amount of non-trivial choiches like wich framework theory we should use, how to formalize the concept of infinity, potential or actual infinity? Cardinals or ordinals? How we define them? Axiom of choiche or not?

This mess of possible choices and interpretation makes your question blurry and ill defined even if I admit that most of the interpretations we can give to the question are really interesting.

Quote:Do Doner and Tarski give a different answer to 2^^w ?
Sure, lets define first the ordinal tetration for limit ordinals
\( {}^\lambda\alpha=sup\{{}^\delta \alpha:\delta<\lambda\}=sup\{0;\alpha;{}^2\alpha;{}^3\alpha;...\} \)

So \( {}^\omega 2=sup \{{}^n 2:n<\omega\}=sup\{0,2;4;16;...\}=\omega \)

About the first fixed point of the function \( \alpha \mapsto \omega ^{\alpha} \)
or in other words the solution of \( \omega ^x=x \) we have that \( {}^\omega \omega \) is the first fixed point and is called epsilon zero

\( \epsilon_0={}^\omega\omega=sup\{{}^n \omega:n<\omega\}=sup\{0,\omega,\omega^{\omega},{}^3\omega,{}^4\omega, ...\} \)

The next fixed points can be defined recursively as follows for all the ordinals

\( \epsilon_{\delta+1}={}^\omega\epsilon_{\delta}=sup\{{}^n \epsilon_{\delta}:n<\omega\}=
sup\{ 0, \epsilon_{\delta}, \epsilon_{\delta}^{\epsilon_{\delta}}, {}^3\epsilon_{\delta}, {}^4 \epsilon_{\delta},...\} \)
for limit ordinals we have
\( \epsilon_{\lambda}=sup\{\epsilon_{\alpha}:\alpha<\lambda\} \)

Is possible to see that for the recursive definition we do not iterate the ordinal superexponentiation \( x \mapsto {}^x \alpha \) but we use the iteration of the map \( x \mapsto {}^\alpha x \) so the ordinal pentation is not involved at all with the fixed points.
Quote:I still challenge to give exp^[1/2](w) even under the assumption that ZFC is good.
In my opinion the theory of surreal numbers can help us to extend the iteration of ordinal operations beyond the set theoretic ordinals (that are discrete, integer-like).

But is hard to think about the iteration of the Cardinal operations... really hard.
If I remember good the cardinals number are not even linear orderable without the axiom of choiche and the problem is that for infinite cardinals the cardinal exponentiation is linked more with the combinatorial constructions (like set of maps, powersets) rather than the recursive definitions.

So the key could be the combinatoral meaning of tetration (like the author of the questions seems to suggest).

Thinking about something simple I noteced that the beth numbers (I have to use the mathfrak symbol) are defined using iterated cardinal exponentiation.

\( \mathfrak B _0:=\aleph_0=\omega \)

\( \mathfrak B _{\alpha+1}:=2^{\aleph_0} \)

\( \mathfrak B _{\lambda}:=sup\{\mathfrak B _{\alpha}:\alpha<\lambda\} \)

the exponentiation is not the ordinal one, note that the cardinal exponentiation push us beyond the countable while the ordinal one (with the epsilon fixed point) can't go beyon the first uncontable \( \omega_1=\aleph_1 \)

Anyways we can define a kind of tetration that is defined for the cardinal base \( \aleph_0 \) and for transfinite ordinal superexponents

\( {}^1 \aleph_0=\mathfrak B _0=\aleph_0 \)

\( {}^2 \aleph_0=\aleph_0^{\aleph_0}=2^{\aleph_0}=\mathfrak B _1 \)

\( {}^{\alpha+1} \aleph_0=\aleph_0^{\mathfrak B _{\alpha}}=2^{\mathfrak B _{\alpha}}=\mathfrak B _{\alpha} \)

But about beth omega... that should be \( {}^\omega \aleph_0=\mathfrak B _\omega \) I'm not really sure it is also inaccessible... wikipedia says that it is the smallest strong limit cardinal but ZFC doesn't prove it (so I guess a fixed point of the cardinal exponentation) but there is the regularity requirement which I don't fully understand (it has to do with cofinalty but beth omega is the union of countable many smaller ordinal so i guess it is not inaccessible).

PS: I made i big mistake because strong limit doesn't mean that is a cardinal exp. fixed point but that for every smaller cardinal \( \kappa<{}^\omega \aleph_0 \) then \( 2^\kappa<{}^\omega \aleph_0 \). The fixed point statement is not compatible with the Cantor theorem.

MSE MphLee
Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)
S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)
What I wrote is in perfect harmony with sup and beth numbers.

Maybe I should apologize for my tricky question.

Tricky because trying to answer might be a trap.

Long ago , I was challenged on sci.math to answer my own question.

There cannot be a bijection from exp^[1/2](w) to w nor 2^w.

Here is why :

Let f(x) be exp^[1/2](x) + o(1).

If card f(w) = card w then

card f(f((w)) = card f( card f(w) ) = card f(w) = card(w)

But f(f(w)) = 2^w


If card f(w) = card 2^w then

card f(f(w)) = card f ( card f(w) ) = card f ( 2^w ) = card(2^(2^w))

But f(f(w)) = 2^w


Likewise for exp^[a] with noninteger a.

SO exp^[1/2] does not exist in set theory.

SO set theory is not suitable for functions since only the power function 2^ is a function that exists and makes a "difference".

( Polynomials do not make a "difference" , they do not change cardinality )

Considering that infinite set theory is not suitable for functions , it makes more sense to be skeptical about its use in other branches of math such as number theory , algebra , calculus and dynamical systems.

Also notice that even substraction is not well defined (for ordinals) since w-1 does not even exist.

That should shed some light on my skeptisism.
( it is JUST the tip of the iceberg , there is way more reason to be skeptical )

Since half-iterates (functions) do usually not exist in set theory , I do not see how one can continue to combine set theory and dynamics.



" Truth is what does not go away when you stop believing in it "

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