Then why does Wolfram say
...they are some quantity that is explicitly nonzero and yet smaller in absolute value than any real quantity. — Banno

Because infinitesimals are paradoxical.

...so the paradox is that Wolfram mathworld has a different definition to you.

I like this (a lot). So look here, infinitesimals are zeroish, but not zero! You're geniusish but no, you're not a genius! It's a cheat code in the game! You fool the system with ishyness. :snicker:

No. Infinitesimals are "explicitly not zero".

so the paradox is that Wolfram mathworld has a different definition to you — Banno

Is that a paradox? Don't think so. There has never been a mathematical object as paradoxically as the infinitesimal.

No. Infinitesimals are "explicitly not zero". — Banno

Ok, but they could be described as zeroish. That's the point, oui monsieur?

Rather, adding certain non-logical axioms results in contradictions — TonesInDeepFreeze

Could you give an example of a non-logical axiom and explain what makes it non-logical?

zeroish — Agent Smith

No.

Is that a paradox? — Hillary

No, that's irony.

Still not seeing a paradox.

No, that's irony.

Still not seeing a paradox — Banno

The paradox lies in the irony. :lol:

No, seriously how can you create a non-zero interval out of length pieces that have zero length? By taking infinitely many? How you do that? You might counter they don't have zero length and it merely approaches zero, but that begs the question. If the distance between two points decreases, the distance will get zero and they touch. The paradox is that they never touch while able to break on through.

...how can you create a non-zero interval out of length pieces that have zero length? — Hillary

But infinitesimals do not have zero length. So that's not what is happening.

...how can you create a non-zero interval out of length pieces that have zero length?
— Hillary

But infinitesimals do not have zero length — Banno

I wrote:

You might counter they don't have zero length and it merely approaches zero, but that begs the question — Hillary

No. — Banno

Why?

How would you round off 0.0001?

Infinitesimals are nonzero. Think that was mentioned. That's not begging the question, that's called a definition.

How would you round off 0.0001? — Agent Smith

Life's too short for this. 0.0001 is not an infinitesimal.

that's called a definition — Banno

But the very definition is paradoxical..

Life's too short for this. 0.0001 is not an infinitesimal. — Banno

Answer the question instead of beating around the bush.

If 0.0001 rounds off to 0, a fortiori an infinitesimal rounds off to 0, oui? Zeroish.

But the very definition is paradoxical.. — Hillary

How, exactly?

Answer the question instead of beating around the bush. — Agent Smith

Your question makes no sense.

How, exactly? — Banno

Approaching without being able to touch.

Still not seeing it.

Answer the question instead of beating around the bush. — Agent Smith

:lol:

Exactly, brother Agent!

Still not seeing it. — Banno

Consider two points. There is space between them. For the infinitesimal to exist the can never touch. However close they get.

I have a basic understanding of Continuity. Get on with it. Where's the paradox? Where's the (p & ~ p)?

For those (like me) who aren't mathematicians, a great way to understand this is to look at the history of mathematics and how much great minds have pondered these question throughout the centuries. Then you get more understanding of how the debate has gone as there actually is a historical narrative how humans have thought about these issues and how we have gotten to where we are.

The thing is that a mathematicians or math education can (and usually does) look at this ahistorically. They just give you the end result, at worst basically an algorithm to use with not much debate about the underlying issues. A proof is given and that's all. Once you understands let's say the notion of limits, for some it looks quite meaningless to ponder about the motion paradoxes of Zeno. These are basically foundational questions about mathematics, and then mathematicians (or philosophers) do understand the question better from that viewpoint.

I have a basic understanding of the continuum. Get on with it. Where's the paradox? — Banno

The paradox lies in the break-up of the continuum in points. If you get on with it a little harder you might see it. The distance between two points gets smaller and smaller. But there will always remain space between them. You can say it's defined like that, but it's non-coherent.

The SEP article on continuity and infinitesimals is set out in an historical order. It gets complicated. Ah dinnae ken. There might be some stuff in it relevant to the topic, but it's not obvious.

The paradox lies in the break-up of the continuum in points. If you get on with it a little harder you might see it. The distance between two points gets smaller and smaller. But there will always remain space between them. You can say it's defined like that, but it's non-coherent. — Hillary

There's nothing paradoxical in that. It's just odd, not paradoxical. A topic for investigation, not @Agent Smith's broken maths or broken logic.

The very notion of laying infinite infinitely small intervals together is nonsensical. Like the other way round, zero infinite big intervals.

There's nothing paradoxical in that. — Banno

A paradox goes against established opinion.

The very notion of laying infinite infinitely small intervals together is nonsensical. — Hillary

Why?

The maths works. What more is there?

A paradox goes against established opinion. — Hillary

But infinitesimals are established opinion.

But infinitesimals are established opinion. — Banno

Yes, but the majority of people think the opposite. That establishes the paradox. The twin paradox goes against established opinion (no time delay). Differentials idem dito.