Opinions on infinitely close things?

[Radarithm]

What are your opinions on infinitely close objects / numbers?
I believe in them, because I can say 3.9 is close to four, but 3.999 is as well, and so is 3.99999999.
I've heard of LOTS of scientists who disagree with this, can anyone prove this wrong (or right)?

 

[CompuChip]

I would agree that 3.9 is close to four, and 3.999 is closer to it even, but no finite amount of 9's will make it equal to four.

I suggest reading https://www.physicsforums.com/showthread.php?t=507001 first, then we'll talk :)

 

[jbunniii]

The distance between two nonequal real numbers is a positive real number. Since there is no smallest positive real number, there is no number $x$ "infinitely close to" $y$ with $x \neq y$.

Proof that there is no smallest positive real number: you supply your candidate $x > 0$, I'll respond with $x/2$.

 

[economicsnerd]

It depends on how you define "number".

 

[CellCoree]

As a scientist, my opinion on infinitely close objects or numbers is that they exist in theory but may not necessarily exist in reality. In mathematics, the concept of infinitesimally small quantities is used to solve certain problems and equations, but in the physical world, there is a limit to how small something can be. For example, in quantum mechanics, the Planck length is considered to be the smallest possible length in the universe.

However, in some cases, the concept of infinitely close numbers can be useful in approximations and calculations. For example, when dealing with limits in calculus, we use the concept of approaching a value infinitely close to a certain point.

Ultimately, whether or not infinitely close objects or numbers exist is a philosophical and mathematical debate. While some scientists may believe in their existence, others may argue that they are simply a theoretical concept. It is important to consider the context and application of these concepts in order to determine their validity.