in a recent MO question, link, discussing the current foundations of mathematics, the author linked a video lecture by Prof. Voevodsky, which argues against the principle of $\epsilon_{0}$-induction used in Gentzen's proof of the consistency of PA.

In discussions arising from the question, some people commented that imagining an infinite descending chain in $\epsilon_{0}$ is "crazy".

I would like to understand better this ordinal, since I actually don't know exactly how to depict it in my mind.

I have clear in my mind the order associated with the finite ordinals. I use in my mind a notation of the following kind:

$1 = I$

$2= II$

$3= III$

$4= IIII$

$\omega = (III\dots)$

$\omega+1= (III\dots)I$

$\omega +2 = (III\dots)II$

$\omega + \omega= \omega \cdot 2= (III\dots)(III\dots)$

In general I understand $\alpha + \beta$ as the juxtaposition of the two representations.

$\omega\cdot 3 = (III\dots)(III\dots)(III\dots)$

$\omega\cdot \omega = \omega^{2} = \big( (III\dots)(III\dots)(III\dots)\dots\big)$

In general I understand $\alpha \cdot \beta$, by replacing each $I$ symbol in $\beta$ with the representation of $\alpha$. So

$\omega^{3}=\omega^{2}\cdot \omega = \big( \omega^{2} \omega^{2} \omega^{2} \dots \big)$

This allows me to *visualize* every ordinal of the form $\omega^{n}\cdot m + k$, with $n,m,k$ naturals (i.e finite ordinals). So far I have absolutely no doubt that there are no infinite descending chain in ordinals of the form $\omega^{n}\cdot m + k$.

However I start having problem with the ordinal $\omega^{\omega}= \bigsqcup_{n<\omega}\omega^{n}$. Do you have any idea on how to visualize $\omega^{\omega}$ is a way consistent with the representation used above (which i actually found here) ?

Anyway, looking at wikipedia, I still manage to visualize $\omega^{\omega}$ as the set of infinite strings of natural number, having only finitely many digits different from $0$.

Still I have no doubt that there are no infinite descending chain in $\omega^{\omega}$.

Perhaps i might be able to understand $\omega^{\omega^{\omega}}$, namely the set of infinite strings labeled with elements of $\omega^{\omega}$, having only finitely many elements different from $0$. Or (i guess) equivalently a $\omega\times\omega$ square labeled with naturals, where only finitely many columns are different from $0^{\omega}$, and all of these non constant-$0$ columns, contains only finitely many digits different from $0$.

However I do not know how to visualize $\epsilon_{0}$. I mean I know that the elements of $\epsilon_{0}$ can be represented by finite-branching finite trees labeled with natural numbers, but that doesn't give me a strong intuition about the fact that no infinite chain exists, so I guess its not a great picture (or at least I do not understand it properly, yet).

**Questions**

A) Could you suggest a way to visualize $\omega^{\omega^{\omega}}$? It should be in such a way to convince me about the fact that there are no infinite down-chain.

B) Could you suggest a way to visualize $\epsilon_{0}$, again arguing that it should be very clear that there are no infinite down-chain.

C) Could you please state your opinion about Prof. Voevodsky, which argues against the principle of $\epsilon_{0}$-induction used in Gentzen's? This shouldn't be a duplicate of the previous, wider thread link, I'm only interested in this little bit of Voevodsky's talk.

Thank you in advance,

bye

matteo