Is 1 Equal to Infinity? A Discussion

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Discussion Overview

The discussion centers around the provocative question of whether 1 can be considered equal to infinity. Participants explore various mathematical and philosophical implications of this idea, including concepts of divisibility, infinitesimals, and the nature of limits. The conversation includes both theoretical considerations and challenges to the validity of the initial proposition.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants express skepticism about the idea that 1 equals infinity, arguing that it is obviously false based on basic numerical comparisons.
  • One participant suggests that 1 is infinitely divisible, proposing that this property might justify equating it with infinity.
  • Another participant questions the meaning of "infinitely divisible" and its implications for the equality of 1 and infinity.
  • There is a discussion about the concept of infinitesimals, with some participants asserting that real numbers cannot be divided into infinitesimal parts, while others argue that the concept of infinitesimals is relevant in certain mathematical contexts.
  • Participants clarify that "infinitesimally small" does not imply the existence of infinitesimal real numbers, but rather refers to arbitrarily small quantities.
  • Some participants emphasize the importance of precise terminology in mathematical discussions, particularly regarding limits and the definition of integrals.
  • There is a contention over whether the limiting size of partitioning pieces in an integral can be considered zero or merely tending towards zero.

Areas of Agreement / Disagreement

Participants generally disagree on the validity of equating 1 with infinity, with multiple competing views on the implications of divisibility and infinitesimals. The discussion remains unresolved, with no consensus reached on the initial proposition or the related mathematical concepts.

Contextual Notes

Participants highlight limitations in their understanding of terms like "infinitesimal" and "infinitesimally small," as well as the need for clarity in mathematical definitions. The conversation also reflects varying interpretations of limits and their application in calculus.

mintparasol
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or 1 = infinity

Discuss
 
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mintparasol said:
or 1 = infinity

Discuss

Why should we discuss it?
 
arildno said:
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
 
mintparasol said:
Maybe others would like to share their thoughts?


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?
 
mintparasol said:
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.

Why does infinite divisibility make 1 equal to infinity?
 
  • #10
Any real number can be divided up into infinitesimally small pieces, but that doesn't mean that the number is infinitely large.
 
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  • #11
Mark44 said:
Any real number can be divided up into infinitessimally small pieces

Why would you say that? There are no infinitesimal real numbers.
 
  • #12
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.
 
  • #13
Jarle said:
Why would you say that? There are no infinitesimal real numbers.

He didn't say into infinitesimal real numbers.
 
  • #14
Mentallic said:
He didn't say into infinitesimal real numbers.

He said real numbers can be divided up in infinitesimally small pieces, implying infinitesimal real numbers.

Mark44 said:
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.

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).
 
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  • #15
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
 
  • #16
Jarle said:
He said real numbers can be divided up in infinitesimally small pieces, implying infinitesimal real numbers.

Aren't you satisfied with the idea of limiting the size of each piece to infinitesimally small?
 
  • #17
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:

http://en.wikipedia.org/wiki/Infinitesimally_small

so I'm not quite sure how to interpret infinitesimally small.
 
  • #18
Mentallic said:
Aren't you satisfied with the idea of limiting the size of each piece to 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.
 
  • #19
Jarle said:
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 \infty=1
 
  • #20
Mentallic said:
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.

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.
 
  • #21
This is silly.
 

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