Does 0.999... Exist? Evidence & Argument

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SUMMARY

The discussion centers on the existence of the number 0.999..., which is analytically equal to 1.0. A key argument presented suggests that if 0.999... does not equal 1, it would imply that it exists independently on the real-number line, which leads to contradictions regarding the properties of the semi-open interval [0,1). The conclusion drawn is that 0.999... lacks a definitive position on the real-number line, relegating its existence to metaphysical considerations. However, it is also established as a fact that 0.999... is equal to 1, affirming its existence.

PREREQUISITES
  • Understanding of real numbers and their properties
  • Familiarity with the concept of limits in calculus
  • Knowledge of the semi-open interval [0,1)
  • Basic grasp of mathematical proofs and arguments
NEXT STEPS
  • Explore the concept of limits in calculus to understand the relationship between 0.999... and 1.0
  • Research the properties of the real-number line and the implications of continuity
  • Investigate the philosophical implications of mathematical existence and metaphysics
  • Review mathematical proofs that demonstrate the equality of 0.999... and 1
USEFUL FOR

Mathematicians, philosophy students, educators, and anyone interested in the foundations of mathematics and the nature of numerical existence.

Opus131
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We know that analytically 0.999... = 1.0. But does 0.999... have any sort of "independent" existence (on the real-number line or otherwise)? Here's my argument against that:

Consider the semi-open line segment [0,1). This set of points has no greatest member. Let's say it did have a greatest member. Then surely that would equal 0.999... Put another way, if 0.999... doesn't equal 1, then we have to conclude that 0.999... only equals 0.999..., which is to say that it's in a class by itself. But where on the real-number line is it located? Intuitively, it would have to be the point "just before" unity. But no such point exists because no two points are "next to" each other on the line. Therefore the idea that 0.999... exists is vague and has to be relegated to the realm of metaphysics.

My argument is intuitive, not analytical. But is the conclusion correct?
 
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Please see the FAQ thread about this in the General Math forum:

https://www.physicsforums.com/showthread.php?t=507002

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Last edited by a moderator:
I'm not sure what you mean by "analytically 0.999... = 1.0". It is a fact that 0.999... is equal to 1 and since 1 exists, yes, 0.999... exists.
 

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