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0 pointsyequalsx9y ago0 comments

Property of real numbers: between distinct real numbers is at least one other number. Now try to find a decimal representation of a number bigger than 0.9999999... but less than 1.0. You clearly can't. They must be equal. No need for infinitesimals.

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ryanmonroe9y ago

Okay, so it follows from completeness. Now prove that real numbers are complete (or provide a construction of the reals that uses completeness as an axiom) without using concepts foreign or confusing to someone with a middle school level exposure to math.

yequalsxOP9y ago

It depends on where you want to start your axioms. We can go the Whitehead/Russell route or just use this as an axiom.

We convince children that 1+1 =2 without delving into the Peano axioms. It's ok to not delve too deeply into the axiomatic structure of the reals.

nzp9y ago

You're just proving their point. None of the things you talk about are even remotely obvious or “natural” to people we're talking about. You want them to ”choose axioms”? Axiom-a-whaaa? 1+1=2 does not need Peano axioms because it's cognitively fundamentally different than 0.999...=1.0. It is immediately obvious because dealing with simple arithmetic on natural numbers is in everybody's experience constantly. In other words we do not need to convince them. Clever untrained people would be able to get 0.999... thing quickly if you gave them some explanation and let them think a bit, if they cared, but properties of real numbers are far from obvious or basic.

JadeNB9y ago

> Okay, so it follows from completeness.

There is nothing to do with completeness here; 0.999… = 1 is a statement about a series of rational numbers summing to a rational number, and the convergence of the series is established by the fact that it sums to the right-hand side, not by an abstract appeal to completeness.

soberhoff9y ago

I'd argue that the distance is 0.00000... = 0 and that distance = 0 implies that the points are equal.

NhanH9y ago

Doesn't that property come from the fact that there is no infinitesimal in real number?

kazinator9y ago

> find a decimal representation of a number

0.99999999...1

This denotes the abstract idea that we take 0.999... (infinite number of 9's) and add another digit.

This is no more or less abstract than 0.999... to begin with.

0.999... is a unicorn, and 0.999...9 is a unicorn with a pink ribbon on the tip of its horn.

lisper9y ago

> add another digit

But you can't. All of the places where you might want to add a digit are already occupied by 9's.

kazinator9y ago

So you're saying a countable infinity can be occupied, so that there is no room for one more.

You must then disbelieve concepts such as that the even integers can be put in 1:1 correspondence with all integers; i.e. that there are exactly as many even integers as there are integers.

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ryanmonroe9y ago

yequalsxOP9y ago

It depends on where you want to start your axioms. We can go the Whitehead/Russell route or just use this as an axiom.

We convince children that 1+1 =2 without delving into the Peano axioms. It's ok to not delve too deeply into the axiomatic structure of the reals.

nzp9y ago

JadeNB9y ago

> Okay, so it follows from completeness.

soberhoff9y ago

I'd argue that the distance is 0.00000... = 0 and that distance = 0 implies that the points are equal.

NhanH9y ago

Doesn't that property come from the fact that there is no infinitesimal in real number?

kazinator9y ago

> find a decimal representation of a number

0.99999999...1

This denotes the abstract idea that we take 0.999... (infinite number of 9's) and add another digit.

This is no more or less abstract than 0.999... to begin with.

0.999... is a unicorn, and 0.999...9 is a unicorn with a pink ribbon on the tip of its horn.

lisper9y ago

> add another digit

But you can't. All of the places where you might want to add a digit are already occupied by 9's.

kazinator9y ago

So you're saying a countable infinity can be occupied, so that there is no room for one more.

You must then disbelieve concepts such as that the even integers can be put in 1:1 correspondence with all integers; i.e. that there are exactly as many even integers as there are integers.

1 more reply

j / k navigate · click thread line to collapse