[QUOTE=retina;275520]Perhaps you could explain what you are referring to?[/QUOTE]
I don't know to what you refer.
(Brief Encounter)

[QUOTE=retina;275520]I've no flash player here. Perhaps you could explain what you are referring to?[/QUOTE]

As exemplified in a scene from Toy Story 2: Stationary pylons create turbulence to Laminar flow. Moving pylons create chaos?

[QUOTE=imwithid;275535]As exemplified in a scene from Toy Story 2: Stationary pylons create turbulence to Laminar flow. Moving pylons create chaos?[/QUOTE]

More than that: Only by a great bit of good fortune do the toys make it across the street in one piece, and they unknowingly leave a great deal of chaos behind them.... so it goes when black is confused with white....or zero with infinity...remembering that the isomorphism breaks down when addition or subtraction is considered.

The video is fun to watch....think of Charlie Chaplin in "Safety Last" for the inspiration used....

A few questions, and a possible retort !
 

[QUOTE=Christenson;275471]Let me re-phrase: (I've been working problems out of Leveque's number theory book, but didn't quite get my proper mathematical hat on....)
Under the map R*-->1/R*, where R* is the real numbers with infinity added,
the operations of multiplication and division are preserved, with zero mapped to infinity and vice-versa.
This map is 1 to 1 and onto. and therefore is an isomorphism.

What did I miss? Surely isomorphisms aren't limited to rings, though they may be more useful there...[/QUOTE]

Is that LeVeque's "Topics in Number Theory, Volume One"? This does not seem to be covered in " Elementary Theory of Numbers " ( Dover edition, 1992 ).
I sense that Cantor and Dedekind are lurking nearby. If I understand what you are writing, does that mean I have an IQ > 50. If so, should I truly sell as previously advised?

Help !

[QUOTE=9021951;275451]I've decided to share my Monday musings with a few more people other than myself, so here I come !

Be careful here, there is a trapdoor ![/QUOTE]

Did everyone choose to ignore the paradox ? A trapdoor <> a trapdoor, if one knows that it exists. It can only exist if one is starting to fall through it, and with no knowledge of its existence until that time or later.

Sheesh !

PS I told you that there was a trapdoor !
PPS Why do I keep spinning in circles ?

[QUOTE=9021951;275558]A trapdoor <> a trapdoor, if one knows that it exists. It can only exist if one is starting to fall through it, and with no knowledge of its existence until that time or later.[/QUOTE]So if we follow your "logic". Once we begin our fall through the trapdoor -> then we have knowledge of it -> and then it ceases to exist (because of our knowledge of it) -> and therefore we are no longer falling through a trapdoor (since it doesn't exist). So what are we falling though again?

So What Gives ?
 

[QUOTE=retina;275562]So if we follow your "logic". Once we begin our fall through the trapdoor -> then we have knowledge of it -> and then it ceases to exist (because of our knowledge of it) -> and therefore we are no longer falling through a trapdoor (since it doesn't exist). So what are we falling though again?[/QUOTE]

This is turning into the quagmire that I expected to exist, before I fell into it.

It is the action of falling that explains the trapdoor, because there seemed to be nothing suspicious about proceeding from point A to point B. However, if I am told that there is one trapdoor which may be encountered during the movement between point A to point B, being located at point C, the trapdoor may as well not exist, because I will avoid it by not traveling toward point B via point C. Or I will test each potential step before making it.

I do not know however what it takes to trigger the trapdoor at point C, that not being given. If I know the trigger but not the location, I still will not fall through it, especially at point C.

So if one does not mind me saying so, the above logic may not be relevant as an explanation, but at least I value the sharing of it with me.

Probably everything in the world must be subject to one's perception of it, and what one does with that, that of course being a reasonable assumption.

Just because I am dealing with an equation, does not mean a supposed refutation of it by saying " 1 = 2 " or " 2 = 1 ". That is incorrect, even in my understanding of mathematics.

So by me asserting the ZERO = INFINITY may just mean that I have looked at that equation with a little more scrutiny, which I have chosen not to share with someone else yet. And believe it or not, my IQ being < 50 has nothing to do with my scrutiny; it's just a reflection, perhaps, of what one sees when one looks directly, tangentially, into the surface of a mirror, and wonders about the IQ of the person seen there.

I do next want to talk about Cantor and " Aleph 0 " but I will clear this thread of my presence first !

As for Leveque, I'll give you a hint: It's before about page 17, in his book about "elementary theory of numbers".....he spends about a half-page, discussing homomorphisms and isomorphisms, just after he defines a group, a ring, and a field.

And R* (R with the addition of infinity) isn't quite a group under multiplication, since infinity * zero is no better defined than it is anywhere else and therefore the set is not closed under multiplication; you have to take limits to get meaningful answers. (See "disproving boundedness of crank score" thread for an example of failure to appreciate that point)

Be *very* careful crossing the street -- remember everything is a projection, and some projections are deadly.

[QUOTE=R.D. Silverman;275463]Total horsesh*t. This is gibberish.[/QUOTE]

Next question: How close are (or aren't) we to showing that odd perfect numbers don't exist? Or is all that factoring just raising the lower bounds?

[QUOTE=Christenson;275633]Next question: How close are (or aren't) we to showing that odd perfect numbers don't exist? Or is all that factoring just raising the lower bounds?[/QUOTE]

The first question is meaningless. There is no way to measure "how far
away" we are from any unproven conjecture. There is no metric.
A proof may require just one little small idea beyond what we have
now, or it may require totally new ideas. There is no way to know or
measure the gap.

The factoring just raises the bound. It's been raised to 10^1500.
What they hope to accomplish by raising it further eludes me. It does
not aid in a proof. 10^1500 is just as far from oo as 10.

[QUOTE]What they hope to accomplish by raising it further eludes me.[/QUOTE]We've said it before, but we might as well say it again. We hope to get some intuition on the problem.

[QUOTE]10^1500 is just as far from oo as 10.[/QUOTE]But 10^1500 is not just as far from 0 as 10 is.

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On the topic of the thread, if you define something so that it is inconsistent then of course it will equal anything else.

Is perfection a state or a process? If it is a process, then those who are perfect (i.e. working towards betterment) must simultaneously be imperfect (because progress is happening).