Actual infinitesimal, actual infinity

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Discussion Overview

The discussion revolves around the concepts of actual infinitesimals and actual infinities, particularly in the context of number systems such as the real numbers and non-standard analysis. Participants explore definitions, implications, and the existence of these concepts within various mathematical frameworks.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants question whether there exists an actual infinitesimal analogous to actual infinity, suggesting that zero might fulfill that role.
  • One participant proposes a number system that includes an infinitesimal symbol \epsilon, defined such that \epsilon^2 = 0, allowing for operations with real numbers.
  • Another participant emphasizes that in the standard set of real numbers, there are no infinitesimals or actual infinities, aligning with the Archimedean property.
  • Definitions of "actual infinity" and "infinitesimal" are discussed, with some participants asserting that the real numbers do not contain these concepts as they are traditionally understood.
  • There is a distinction made between actual infinity as a number that cannot be added to and potential infinity as a concept related to functions.
  • One participant references a common definition of Archimedean systems, stating that they contain no infinitesimals other than zero.

Areas of Agreement / Disagreement

Participants express differing views on the existence and definitions of actual infinitesimals and infinities. There is no consensus reached, as various interpretations and definitions are presented without resolution.

Contextual Notes

Definitions of infinitesimals and infinities vary among participants, and the discussion highlights the dependence on specific mathematical frameworks, such as non-standard analysis versus standard real number systems.

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Is there an actual infinitesimal in the way that there is an actual infinity. Or would zero fill that role.
 
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You can define a number system by adding an additional symbol \epsilon and defining it by \epsilon ^2 = 0 and then taking the set of all a+b \epsilon where a and b are real numbers. You can add and multiply like normal using distributivity and commutativity. But in the standard set of real numbers there is no infinitesimal, just like there is no actual infinity
 
What do YOU mean by "actual infinity"? "Non-standard analysis" uses the "hyper-real numbers" with infinitesmals. But, as Office Shredder said, there is no "actual infinitesmal" just as there is no "actual infinity".
 
Office_Shredder said:
But in the standard set of real numbers there is no infinitesimal, just like there is no actual infinity

I thought 0 was an infinitesimal.
 
No, it isn't.

"In common speech, an infinitesimal object is an object which is smaller than any feasible measurement, hence not zero size, but so small that it cannot be distinguished from zero by any available means."

" number system is said to be Archimedean if it contains no infinite or infinitesimal members."
and, of course, the real numbers are Archimedan.

http://en.wikipedia.org/wiki/Infinitesimal
 
I'm accustomed to the definition "a system is Archimedian iff the only infinitesimal it contains is zero".
 
HallsofIvy said:
What do YOU mean by "actual infinity"?

The number that can't be added to.

I'm understand a potential infinity to be more like a function.
 
Pjpic said:
The number that can't be added to.

I'm understand a potential infinity to be more like a function.

If by infinity, you mean "a number which no other is greater", then the real numbers contain no infinities.

If by infinitesimal, you mean "a nonzero number which is less in magnitude than all others", then again, there are no infinitesimals in the reals.
 
Pjpic said:
HallsofIvy said:
What do YOU mean by "actual infinity"?

The number that can't be added to.

I'm understand a potential infinity to be more like a function.

Pjpic said:
Is there an actual infinitesimal in the way that there is an actual infinity. Or would zero fill that role.
The reason I asked was that your original post (which I have quoted here) implied that there exists an "actual infinity". There does not- not in the real numbers. There are many different ways to define both "infinity" and "infintesmal" in other systems.
 

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