jgill

Mathematicians? Not necessarily. "Flaws" . . . not necessarily. Incidentally, your compact form of Leibnitz expansion has a simple error. And 1/3 =.33...

So how does Zeno produce a non-measurable set? :chin:

So the AC is a logical consequence of infinite divisibility and thus is not an axiom? :gasp:

I don't know about that, never having gone through the proof of B-T. However . . . Wiki: Unlike most theorems in geometry, the proof of this result de...

In math a point is usually a position dependent upon a framework. Take away the framework, does the point still exist? It seems to since it crops up i...

I hereby revoke my two axioms and my original frivolous statement about integers. Happy Holidays! :nerd:

https://en.wikipedia.org/wiki/Equivalence_class If you are using mathematical symbols, then the above should suffice. If you are arguing why a thing i...

No, no, no ......! You have challenged my axioms. The ball is in your court! :nerd:

More of a concept, actually. An expression "becomes infinite" if it grows without bound. No need for a symbol. But it's there to be used if one wishes...

Hmmm. Now where did we come up with a number system base 10? Are my two axioms inconsistent? :roll:

Equivalence classes. Simple example: 1/3 = E = {n/m: n/m = 1/3}. Reflexive: shows 1/3 belongs to E. Symmetry: 1/3 = 2/6 and 2/6 = 1/3. Transitive: 1/3...

X\ne X See Banno's Game.

The Axiom of No Choice: For any collection of non-empty sets there is at least one way to avoid choosing an element from each set. (This will lead to ...

The sum of any two integers is zero.

It's a little more complicated than that. But go ahead and make one up. Should be fun. :smile:

Lots of changes coming, shifting agricultural patterns, beachfront properties inundated, and for those in the UK, get your woolies out when the Gulf S...

It's hard to argue with that. :chin:

The Axiom of Choice sounds so very benign (one can pick an element out of each non-empty set in a collection of sets), but look what it leads to! And ...

How about the real plane where a point is denoted by (x,y) ? Still have a problem? Complex analysis may be hard to justify for those who know little m...

What about a "point" in N-dimensional space? There is no problem if a point is in essence an N-tuple of real numbers. Of course, for those who do not ...

OK. Please describe those contradictions. I've not encountered them in my many years. :roll:

All you have to do is come up with an example showing this to be the case, rather than argue in an abstract way about it. Maybe you have, as I haven't...

As a math guy I might question "true science" , but I am not a philosopher so am biased. The only language games in which I indulge are crossword puzz...

Wiki: incommensurable generally refers to things that are unlike and incompatible, sharing no common ground (as in the "incommensurable theories" of t...

With what?

Most working mathematicians do not feel mortally obliged to prove the existence of an object by producing the object. It is best if they can, however....

Very well put. :cheer:

The news media has become very selective in what it states and prints and is politically biased - on both ends of the political spectrum. How are phil...

Well, software, being of a modern generation of mathematicians, has opened my eyes again: Wiki:"Other mathematical systems exist which include infinit...

One-to-one and onto was not so difficult to say. But your generation learned differently. And it's the coin of the realm, now. :nerd:

Beyond finite instances, cardinalities are not numbers. They are equivalence classes.

I don't know. But to some extent scientists - especially Greek - were sheltered by Muslim societies during the Dark Ages. Mathematical contributions a...

I suppose religion is not PC, what with "Supreme Being" and all. It's a language minefield out there. :gasp:

The basic theory of probability and statistics are pretty well established, but when it comes to practical applications there are some problems. Remem...

Mathematical set theory and foundations provide a playground for combative metaphysicians. And if you contemplate my question you might see why.

Talk about thread-drift. I don't like the expression "quantum supremacy" because it implies no further scientific progress is likely in computing. Not...

Cardinality has to do with sets being in a 1:1 correspondence with each other. \aleph_0 is not a number in the common sense of the word. It denotes an...

Really? Consider N={1,2,3,...} and S={1,9} No 1:1 correspondence. You are missing "equivalence" "A" proper subset? Not "The", I think.

Language frames much of our thinking, placing thoughts in a context that allows communication of those thoughts, sometimes imperfectly, to others. If ...

And yet it sounds so simple. A set is a collection of "elements" or objects and the axiom says we can consider the collection itself as an object, but...

When you get down to the nitty gritty and use numbers, as in a computer program, they are all rational and thus countable. So how many rational number...

I don't think measure theory has much to do with this. But maybe it does. A non-measurable set is not a triviality. A point on the real line simply co...

It's interesting that the axiom of infinity and the axiom of choice can lead to such lengthy discussions. On the surface they seem so benign. The firs...

www128.pair.com/r3d4k7/Mathematicae7.html https://en.wikipedia.org/wiki/Riemann_sphere#Extended_complex_numbers You can't even do links here it seems....

As a professional math guy, here is my opinion regarding infinity and the complex plane: For me and my colleagues, |z| getting larger and larger witho...

OK. It requires $5/month to do so.

OK. Got the message. $5/month for permission to hot-link images. Maybe, maybe not. We'll see if it's worth it. :sad:

Carry on in your crusade. It's above my pay grade and below my interest. (that's not my quote, someone on Wiki TALK). The link to math stackexchange i...

I can't get the image to stay. Suggestions?

I've never known a mathematician who actually used z/0 = infinity. It's simply a symbol that ultimately refers to the north pole of the Riemann sphere...