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October 6th, 2018, 01:01 PM   #81
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I am trying to follow your post here.

Originally Posted by sevensixtwo View Post
Consider the definition of a real number: it is a cut in the real number line. For any real radius around the origin, the real number line leaves that radius.
Ok, so if $x$ is a real radius that simply means $x$ is a real number, correct?:

$$x \in (0,\infty) \implies \exists y \in \mathbb{R} \text{ such that } y > x$$

Originally Posted by sevensixtwo View Post
Cuts in the part beyond the radius are real numbers in the neighborhood of infinity.
Ok, so:

$$y > x \implies y \in \text{ 'the neighborhood of infinity'}$$
$$x = 1 \implies (1,\infty) = \text{ 'the neighborhood of infinity'}$$

Originally Posted by sevensixtwo View Post
I have defined these types of real numbers to have the form "infinity minus b."
This seems nonstandard. I believe the standard would be that $\infty - b = \infty \neq y \in (1,\infty)$.

Originally Posted by sevensixtwo View Post
Since it is known that one can make definitions, I have proven that a real number can have this form.
The real numbers are already defined. One can define a number in any fashion they see fit, but that doesn't make it a real number. Have you proven that $\infty - b \neq \infty$ and instead that $\infty - b \in \mathbb{R}$ or something? What am I missing...?
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