Isn't the definition of infinitesimal contradictory?

[wtf]

Could you please elaborate further and define infinitesimal?

An infinitesimal is a quantity that is:

* Strictly greater than zero; and

* Less than 1/n for any positive integer n.

This is the actual mathematical definition of an infinitesimal.

With this definition it's easy to prove that there are no infinitesimals in the reals; and that there ARE infinitesimals in the hyperreals.

[bahman]

Could you please elaborate further and define infinitesimal?

An infinitesimal is a quantity that is:

* Strictly greater than zero; and


* Less than 1/n for any positive integer n.


This is the actual mathematical definition of an infinitesimal.

With this definition it's easy to prove that there are no infinitesimals in the reals; and that there ARE infinitesimals in the hyperreals.

Isn't the bold part a definition for smallest number?

[wtf]

Isn't the bold part a definition for smallest number?

No. If ε > 0 then ε > ε/2 > 0 so ε is not the smallest positive real number. You yourself made essentially the same point in your OP, didn't you?

In the hyperreals or in any system containing infinitesimals, there is no smallest infinitesimal for the same reason. You can always divide by 2.

[bahman]

No. If ε > 0 then ε > ε/2 > 0 so ε is not the smallest positive real number.

If it is not the definition of something, smallest number for example, then it is definition of what?

You yourself made essentially the same point in your OP, didn't you?

Yes. I was exactly pointing to contradiction which is unavoidable when you try to define infintesimal.

In the hyperreals or in any system containing infinitesimals, there is no smallest infinitesimal for the same reason. You can always divide by 2.

So the hyperreals is incorrect unless you can provide a definition for infinitesimal which is not contradictory.

[wtf]

If it is not the definition of something, smallest number for example, then it is definition of what?

It's the definition of an infinitesimal.

Yes. I was exactly pointing to contradiction which is unavoidable when you try to define infintesimal.

There's no contradiction. You are claiming an infinitesimal is the smallest quantity greater than zero, but there is very obviously no such thing. The only contradiction is in your own false claim.

So the hyperreals is incorrect unless you can provide a definition for infinitesimal which is not contradictory.

I did.

[bahman]

If it is not the definition of something, smallest number for example, then it is definition of what?

It's the definition of an infinitesimal.

Yes. I was exactly pointing to contradiction which is unavoidable when you try to define infintesimal.

There's no contradiction. You are claiming an infinitesimal is the smallest quantity greater than zero, but there is very obviously no such thing. The only contradiction is in your own false claim.

So the hyperreals is incorrect unless you can provide a definition for infinitesimal which is not contradictory.

I did.

As far as I understand the star shows remark, a part of your definition which indicates a property. I am asking whether this property is equal to the smallest, largest, etc. To elaborate, you in one hand put a lower limit for infinitesimal, bigger than zero. Therefore you need a higher limit for it too which is defined by, less than 1/n for any positive integer n. I am arguing that the second property (bold part) cannot exist and it is the definition of smallest number since you need a higher limit too.

[-1-]

Could you please elaborate further and define infinitesimal?

I don't know if you are a religious nut, but I give you the benefit of the doubt that you are not.

"Infinitesimal is an amount equal to the amount of how much the religious nuts on this forum are able to comprehend speech, writing, thought, and how much reasoning ability they possess. "

There, done.

[bahman]

Could you please elaborate further and define infinitesimal?

I don't know if you are a religious nut, but I give you the benefit of the doubt that you are not.

"Infinitesimal is an amount equal to the amount of how much the religious nuts on this forum are able to comprehend speech, writing, thought, and how much reasoning ability they possess. "

There, done.

Is that your definition?

[-1-]

Is that your definition?

May I know why you're asking that?

[wtf]

As far as I understand the star shows remark, a part of your definition which indicates a property. I am asking whether this property is equal to the smallest, largest, etc.

There is no smallest positive real number in any model of the reals, for the simple reason that you already gave in your OP. You can always divide a positive number by 2 to get a smaller one.

To elaborate, you in one hand put a lower limit for infinitesimal, bigger than zero. Therefore you need a higher limit for it too which is defined by,

Why? The positive real numbers themselves have a lower limit, zero, but no upper limit.

less than 1/n for any positive integer n. I am arguing that the second property (bold part) cannot exist and it is the definition of smallest number since you need a higher limit too.

Nonsense. Consider the positive reals. They have a lower limit but no upper limit.

[bahman]

Is that your definition?

May I know why you're asking that?

You haven't add much yet. What I am looking for is a proper definition of infinitesimal if that really exist.

[bahman]

As far as I understand the star shows remark, a part of your definition which indicates a property. I am asking whether this property is equal to the smallest, largest, etc.

There is no smallest positive real number in any model of the reals, for the simple reason that you already gave in your OP. You can always divide a positive number by 2 to get a smaller one.

That I am aware of it.

To elaborate, you in one hand put a lower limit for infinitesimal, bigger than zero. Therefore you need a higher limit for it too which is defined by,

Why? The positive real numbers themselves have a lower limit, zero, but no upper limit.

Because otherwise it could be any real number bigger than zero.

less than 1/n for any positive integer n. I am arguing that the second property (bold part) cannot exist and it is the definition of smallest number since you need a higher limit too.

Nonsense. Consider the positive reals. They have a lower limit but no upper limit.

It makes sense.

[-1-]

Is that your definition?

May I know why you're asking that?

You haven't add much yet. What I am looking for is a proper definition of infinitesimal if that really exist.

1. Under what authority do you feel you have the power to decide whether a definition is proper or not?

2. "if (the infinitesimal) really exists." Do you want a definition to something you have trouble even knowing if it exists or not?

Is it the definition of "infinitesimal" you need, or a tangible description of a vague concept that you have for "infinitesimal"?

[A_Seagull]

Infinitesimals don't exist.

Not even numbers exist.

All you have in mathematics are symbols and concepts that are associated with those symbols.

So the symbol '=' is no more than a symbol within the system of mathematics. And that symbol is typically associated with the concept of "equal" and all that entails.

[Atla]

By assuming a weak form of the axiom of choice, one can show the existence of a gadget called a nonprincipal ultrafilter, which one can then use to produce a field (a mathematical system in which we can add, subtract, multiply, and divide) in which there are infinitesimals. This field is called the hyperreals. [Technically there are many such fields since different nonprincipal ultrafilters give rise to nonisomorphic fields of hyperreals. And it's a mathematical curiosity that the Continuum Hypothesis implies that there is only one unique field of hyperreals. As you can see we are in deep foundational waters].

Pardon me for my ignorant question (ignore it if it's just word salad): if we assume the Continuum hypothesis to be correct and so there is only one unique field of hyperreals, then would this uniqueness be somewhat similar to for example the uniqueness of the Monster group from group theory: similar in the sense that it would be a certain specific mathematical structure, found in higher mathematics, that's simply "there", for some reason?

(I just find it fascinating when mathematics seems to discover structures that are simply "there". I'm coming from the physical multiverse hypothesis angle; I have the impression that such structures may be somehow linked to the unique topology of our universe, but in ways we can't really understand yet. I also read somewhere that "set multiverse" ideas are a thing now in mathematics, and people who hold such views are more likely to think the the Continuum hypothesis may be correct.)