- #1

Agent Smith

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- TL;DR Summary
- Defining the Infinitesimal

Cobbling together a definition of

The infinitesimal ##d## is the positive real number greater than ##0## but less than

My problem is how to express the above in logical notation. Some of my attempts follow:

1. Domain ##\mathbb{R}^+##

##\{d : x \geq d\}##

In the domain of positive reals ##d## (the infinitesimal) is the positive real such that all positive reals are either equal to ##d## or greater than ##d##

2. ##\forall x \exists y ((x, y \in \mathbb{R}^+ \wedge x \ne y) \to y < x)##

For all x there exists a y such that if x and y are positive reals and x is not equal to y then y is less than x Here y is ##d##

3. ##d \in \mathbb{R}^+ \wedge \forall x ((x \in \mathbb{R}^+ \wedge x \ne d) \to d < x)##

##d## is a positive real AND for all x such that x is a positive real not equal to ##d## then d is less than to x.

Are all the above correct definitions for the infinitesimal ##d##?

*the infinitesimal*from bits and pieces of info gathered from books and the internet:The infinitesimal ##d## is the positive real number greater than ##0## but less than

*any other*positive real number.My problem is how to express the above in logical notation. Some of my attempts follow:

1. Domain ##\mathbb{R}^+##

##\{d : x \geq d\}##

In the domain of positive reals ##d## (the infinitesimal) is the positive real such that all positive reals are either equal to ##d## or greater than ##d##

2. ##\forall x \exists y ((x, y \in \mathbb{R}^+ \wedge x \ne y) \to y < x)##

For all x there exists a y such that if x and y are positive reals and x is not equal to y then y is less than x Here y is ##d##

3. ##d \in \mathbb{R}^+ \wedge \forall x ((x \in \mathbb{R}^+ \wedge x \ne d) \to d < x)##

##d## is a positive real AND for all x such that x is a positive real not equal to ##d## then d is less than to x.

Are all the above correct definitions for the infinitesimal ##d##?