Y = a^(1/n)….,where “a” is arbitary number

including complex number, while “n” is every positive numbers.
i can’t use the first guest number by 0 (zero).
why does matlab can’t do this?
i think there is some mistake in matlab work. if it is truth that matlab can’t do this, so the conclusion is all the people in the world that used mathlab for numerical solution had make a big mistake…
is anyone can help me if iam wrong?????

In my view, there is no great mystery as to why so little set theory is “needed” for mathematics. The point is that mathematics historically did not start with the study of sets. Set theory, especially set theory as a tool for foundational studies, is a johnny-come-lately that happens to give a very simple and elegant foundation that *suffices* for (essentially) all mathematics. The goal was to find something that would be general enough to do the job, not to find the “lowest bidder” (i.e., the weakest set of axioms that would suffice). Thus we formulate axioms like “every set can be well-ordered” rather than “this set and that set and, oh, by the way, that other set too can be well-ordered” since the latter is pointlessly cumbersome given our aims. Naturally, the result is an extremely powerful axiomatic system.

Only if one has been mis-educated into thinking that mathematics is “really” based on set theory, as opposed to mathematics just being mathematics and just happening to have the property that it *can* be based on set theory, will one be surprised at how strong ZFC is.

On a different note, solrize wondered why Friedman says that math is essentially Pi^0_1 when obviously we have things like P != NP. Friedman’s point is that once we prove some Pi^0_2 or Sigma^0_2 statement, it is almost a conditioned reflex for us to ask immediately for a stronger version of the statement that gives explicit bounds. This drops us down to the Pi^0_1 level. So for example, if someone proves P != NP, we are not likely to be satisfied until we prove something stronger, like “for all n, there is no circuit for SAT having fewer than (1.1)^n gates” or something even stronger than that.

Let Ui be a set of locations of particles of the universe.
U1xU2x …… xUix ….. a set of infinite paths
(Cartesian product).
this set is equal to the void set by the
negation of the axiom of choice.

So there is no more space containing the particles.
The particles collapse on themselves: Big Crunch.
Then Big Bang.

The Big Bang has taken place thus the negation of the axiom
the choice is likely to considered as a good axiom.
Adib Ben Jebara.

All sorts of exceptional objects can be shown to exist, and we can prove that varous universes of discourse are inherently incomplete – but as soon as a particular example is demonstrated, this is done within an inherently countable part of mathematics, and cannot therefore be ‘too pathalogical’. So our discussions of ‘pathalogical objects’ cannot get out of hand. [My Greek teacher once said "no language has a lot of irregular verbs" - don't know enough languages to know whether this is true, but it has an analogue which I think is true of mathematics]

All this is worlds away from ultrafilters.

Another part of the answer to the original question is perhaps that once it is shown [or we believe - possibly wrongly] that our conceptualisations are compatible with set theory, then most of the time it seems easiest to use the concepts developed for a particular part of mathematics than to relate them all the time to sets. This is often convenient, but some of the subtlety and power of set theory may get lost or mislaid on the way.

My original point was that (I suspect) some proofs are easier with ultrafilters than without them, and this may be a case in point. Despite their abstract nature, they can be technically useful, as Terence Tao was also suggesting.

Conjecture: the “general set theory” of George Boolos, augmented with axioms of Infinity and Choice, should suffice as foundations for nearly all algebra and analysis.

It is also conceivable that some sort of second order arithmetic will be become the canonical foundation of mathematics within 30-40 years.

I have learned much from John Burgess’s “Fixing Frege.”