Mowk wrote:What happens when logic and math disagree?

They don't

Math is built on logic - ever heard of Zermelo-Fraenkel set theory?

It's so funny to see another mathematician try to speak a word of sense here

Your input is some nice fresh air, phyllo.

This new \("\frac1{3\times10^\infty}"\) is also pretty funny, as it too must have infinitely more 3s "at the end of endless 0s" in its decimal representation just as all \(\frac1{3\times10^n}\) do.

So either you can always divide a term by 10 one more time, showing \("\frac1{3\times10^\infty}"\) to hardly be some "ending remainder" to endlessness, or the higher order of magnitude 0s really do

never get to these fictitous 3s that never ever appear as "already there" - as would be consistent with the \(\infty\) being erroneously used

AGAIN as a value rather an operator. When will this finally sink in? It will take an infinite amount of time no doubt.

All there really is here is an objection to decimal representations, and as phyllo correctly says, as I too said earlier in the thread, either all math is consistent or it's all broken.

\(\frac9{3}=3\) with absolutely

no problem so either \(\frac{10}3=3.\dot3\) using exactly the same math, or division itself is broken, making math inconsistent and even \(\frac9{3}=3\) invalid...

There's just no consideration of the consequences of trying to make out like there's an end to endlessness for the sake of some illusory remainder....

phyllo wrote:Let's face it :

\(0.\dot01=0\)

Is only logical.

Absolutely correct. You literally never reach the end of the endless 0s to get to that infinitely elusive "1".

Any smallest number will always be inconsistent,

anything you need it to be,

undefined - just the same as \(\infty\), simply because either you could always divide it smaller - or there's an ending limit to its smallness.

This logic in that simple sentence is as exhaustive as it is unequivocally definitive, as well as backing up the consistency of math from its grounding in logical roots, all the way up - for us to be able to have this conversation at all in the first place(!)