or 1 = infinity
Why should we discuss it?
You don't have to if you don't want to. I'm not your papa. It's an interesting concept to me. Maybe others would like to share their thoughts?
Maybe you would like to share your thoughts? Why do you think it's interesting? Why do you think it's reasonable?
At the moment, all I can say is it's obviously not true. 2>1 and 2 is clearly not larger than infinity. I don't understand why you would possibly think this is worthy of discussion
Well, my first thought was "no".
You will need to back up your claims by explaining what you mean by them. On it's own it's pure nonsense. And clearly wrong by any reasonable interpretation.
Maybe I've bitten off more than I can chew here :)
1 is infinitely divisible. It seems to me that in order to work with infinity in math, you should just call it 1.
What is 1 divisible by exactly?
Infinity = 1, but what does god need of a spaceship?
Why does infinite divisibility make 1 equal to infinity?
Any real number can be divided up into infinitesimally small pieces, but that doesn't mean that the number is infinitely large.
Why would you say that? There are no infinitesimal real numbers.
An interval, say [0, 1], of length 1, can be divided into infinitesimally small pieces for calculating the integral of a function that is defined and continuous on that integral. In that sense, a number can be divided into the sum of infinitesimally small numbers.
He didn't say into infinitesimal real numbers.
He said real numbers can be divided up in infinitesimally small pieces, implying infinitesimal real numbers.
No, that is not how the integral is commonly defined. There are no infinitesimals involved in standard analysis. It is usually defined in the form of a limit (the supremum of the lower limit of the approximations by trapezoids).
Standard analysis does not treat infinitesimals at all. Any mention of them should explicitly refer to the extension of the real number field by infinitesimals and infinite numbers as defined in non-standard analysis (which gives an alternative definition of the integral).
infinitesimally small is not the same as an infinitesimal number. It just means arbitrarily small. Similar to how "I can pick natural numbers infinitely large" doesn't mean that there is a natural number that is of value infinity, just that arbitrarily large ones exist
Aren't you satisfied with the idea of limiting the size of each piece to infinitesimally small?
All right, if infinitesimally small commonly refers to arbitrarily small, then I'll agree. I wasn't aware of that. However, wikipedia redirect "infinitesimally small" to "infinitesimal". As you can see:
so I'm not quite sure how to interpret infinitesimally small.
The limiting size of the partitioning pieces is 0 (well, given that the integrand is not linear at any interval), so no - I'm not satisfied with that at all.
I don't believe it is 0, but tends to it. That's the point of limits. Of course while the limiting size tends to zero for each piece, the number of pieces tend to infinite and we've still divided the finite interval into that many pieces. Which is what Mark was getting at in the first place.
There's no need for being picky about terminology or even involve higher mathematical ideas into the mix, because after all, we're simply trying to disprove [tex]\infty=1[/tex]
That a varying size tends to zero means that the limiting size is zero.
I just pointed out that one should not confuse this with infinitesimals. If that wasn't the intention, then I'm guessing we are done here. I'll agree with that we could be cutting some slacks about precise terminology.
Separate names with a comma.