# Smallest positive real number — history of mistake

When I was about 14 (it was long ago by the way) I was sure that there is the smallest number in $\mathbb{R}_{>0}$ (i.e. a real number which is strictly greater than 0). I even somehow knew that the name of this number is $\varepsilon$. Now I know that such number does not exist, this is not the question. But recently I met a guy (he is about 14 now) and he has the same idea of smallest positive number.

I am quite sure that I am not the first one who has done this mistake. People who created analysis probably did this first.

Question 1. Are there famous mathematicians who did this mistake?

Question 2 Is there a standard name for this mistake?

Achilles and the Tortoise is similar but not exactly this.

I am in particular interested in Leibniz Monadology, but it is too vague (at least for me).

Question 3 Did Leibniz have this wrong belief?

Remark 1. The concept of infinitesimal numbers is a less naive one. If you have an infinitesimal number $\epsilon$ then you can consider $\epsilon/2$ (why did not I think about this when I was 14...).

Remark 2. I aware of NSA. But my question is strictly about real numbers (not hyperreal or whatever). I am not asking about people who wanted to formalize the concept.

• Did either of you think this mysterious number is $0.000\cdots001$? – Angina Seng May 5 '18 at 10:11
• It's all about definitions, that not a wrong belief or anything like that... I guess one can try to create some new real numbers with minimal numbers (perhaps by formally defining numbers of the form $0.000\cdots 001$ as @LordSharktheUnknown "suggests" or by formally adding the number $\varepsilon$ and declare it as being the minimal number). I think it is more appropriate to ask why we define the real numbers without such numbers... – Yanko May 5 '18 at 10:18
• @Yanko This question is not about definition. It is about real numbers, mistakes and history of mathematics. – quinque May 5 '18 at 11:14
• At some point in the history mathematicians decided to rigorously define the real numbers. Before that time, it 's hard to say that other things are wrong. (As long as I'm aware one has to use the notion of "limit" in order to properly define the real numbers, so Leibniz has to live before that definition) – Yanko May 5 '18 at 11:29
• Maybe people believe this by defining $\epsilon:=1-0.\bar{9}$. – J.G. May 5 '18 at 12:27

While this does not technically relate to mathematicians believing in a "smallest real number" (- I believe the giants lending us their shoulders to stand on were pretty smart, even in ancient greece... -) the concepts of NSA document that there has been a notion that $\varepsilon$-$\delta$-techniques are somewhat unsatisfactory and "true infinitesimals" should be formalized. So while likely not falling for the same mistake you are describing, mathematicians clearly have been puzzled by related problems.
• Thank you for your remark. By the way, I aware of NSA. I forgot to mention this in my question. But 1. My question is strictly about real numbers. I am not asking about people who wanted to formalize the concept. 2. The concept of infinitesimal numbers is a less naive one. If you have an infinitesimal number $\epsilon$ then you can consider $\epsilon/2$ (why did not I think about this when I was 14...). I will edit my question to make it more clear. – quinque May 5 '18 at 11:04
Within Nelson's Internal Set Theory, you can work within the real number line itself. Here you would choose a fixed nonstandard $\epsilon>0$ and work with a grid of numbers between $-H$ and $H$ where $H=\frac{1}{\epsilon}$. This is enough to do most of calculus, including derivatives and integrals, as well as most of probability theory; see Radically Elementary Probability Theory. This setting provides a satisfactory formalisation for the idea of a "smallest real number", namely $\epsilon$.