Skip to main content

# Tag Info

## Hot answers tagged nonstandard-analysis

35 votes

### Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?

To ask "Is $\frac{dy}{dx}$ a ratio or isn't it?" is like asking "Is $\sqrt 2$ a number or isn't it?" The answer depends on what you mean by "number". $\sqrt 2$ is not an Integer or a Rational number,...
• 491
27 votes

### Basic Geometric intuition, context is undergraduate mathematics

The viewpoint you're groping towards is not crazy, and at least historically you're in excellent company -- e.g., Leibniz had similar ideas when he viewed an integral as a sum of the heights of ...
26 votes

### Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?

There are many answers here, but the simplest seems to be missing. So here it is: Yes, it is a ratio, for exactly the reason that you said in your question.
• 4,804
26 votes
Accepted

### What are hyperreal numbers? (Clarifying an already answered question)

The usual construction of the hyperreal numbers is as sequences of real numbers with respect to an equivalence relation. For example, the real number 7 can be represented as a hyperreal number by the ...
• 5,550
25 votes

### Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?

I am going to join @Jesse Madnick here, and try to interpret $\frac{dy}{dx}$ as a ratio. The idea is: lets interpret $dx$ and $dy$ as functions on $T\mathbb R^2$, as if they were differential forms. ...
25 votes
Accepted

• 19.9k
16 votes
Accepted

### Which branch of mathematics rigorously defines infinitesimals?

Most objections to non-standard analysis seem to be about the use of the axiom of choice in the construction of the field of hyperreals. Non-standard analysis is completely rigorous, but if you're a ...
• 36.4k
16 votes
Accepted

### Nonstandard infinite / hyperfinite sum in IST

Welcome to Math.SE! You did not indicate how much background you have in Internal Set Theory (IST) - but based on what you wrote, it seem like you have a few misconceptions about it. Let us start by ...
• 11.5k
15 votes
Accepted

### Basic Geometric intuition, context is undergraduate mathematics

This is less an answer and more of an extended comment. You seem to be struggling with the idea of a point as contrasted with an infinitesimally thickened point, and it sounds to me like you want to ...
• 10.8k
13 votes

### Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?

The derivate $\frac{dy}{dx}$ is not a ratio, but rather a representation of a ratio within a limit. Similarly, $dx$ is a representation of $\Delta x$ inside a limit with interaction. This interaction ...
12 votes
Accepted

### Dedekind completion of ordered fields

The existence of additive inverses fails in $\mathbb{S}^*.$ A member of $\mathbb{S}^*$ (a Dedekind cut in $\mathbb{S})$ is a subset A of $\mathbb{S}$ such that $A$ has an upper bound, $A$ has no ...
• 10.3k
11 votes

### How do we interpret and perform integrals using infinitesimals?

$\DeclareMathOperator{\st}{st}$I will explain one way to treat your definite integral in the Internal Set Theory (IST) approach to Nonstandard Analysis (NSA). Before beginning, please be aware this ...
• 1,539
10 votes
Accepted

### Can unlimited hypernaturals be represented by increasing sequences?

You certainly can't make $(a_n)$ be strictly monotonic, since that would mean that $a_n\geq a_0+n$ for all $n$. So, for instance, if $k_n=\lfloor\sqrt{n}\rfloor$ we cannot choose any such $a_n$, ...
• 332k
10 votes
Accepted

### Is $e$ transcendental when working with hyperreal numbers?

This is totally wrong. You cannot express $e$ as $(1 + \frac{1}{H})^H$; rather, $(1 + \frac{1}{H})^H$ will be infinitely close to (but NOT equal to) $e$. After all, $(1+\frac{1}{n})^n$ is never ...
• 332k
10 votes

### Cauchy sequence for $0$ in non-standard analysis

In the usual construction of the real numbers from Cauchy sequences, two Cauchy sequences are considered equivalent iff the term-wise difference converges to $0$. In the usual construction of the ...
• 200k
9 votes

### Basic Geometric intuition, context is undergraduate mathematics

I don't know if your post has much to do with "life advice", but the question of whether there should be an "infinitely small but non-zero width" is something that bears answering. The way math is ...
• 226k
9 votes
Accepted

### Hyperreals, other models and 1=0.999....

Let's make our notation more explicit. First, let's briefly recap the standard situation. Decimal representations are really just infinite sums, and in particular 0.9999...:=\sum_{i\in\mathbb{N}}{9\...
• 247k
9 votes
Accepted

### Does 3Blue1Brown's series on Calculus : Essence of Calculus approach it via limits or infinitesimals (or both)?

I'll flesh out @MichaelMorrow's comment, with a qualification: the standard modern approach is to describe everything in terms of limits. Historically, calculus grew out of a desire to understand what ...
• 116k
9 votes

### Nonstandard infinite / hyperfinite sum in IST

You asked several questions that should really be separate posts, but let me start by answering one. You asked: "If anyone could provide a detailed proof that a sum indexed by an unlimited ...
• 43.3k
8 votes
Accepted

### Are there non-standard counterexamples to the Fermat Last Theorem?

I am going to respond to two questions quoted below, which come from this comment. Here "TP" means "transfer principle". I've switched the order of the questions. ... why it does not prove that ...
• 81.8k
8 votes

### Why Cauchy's definition of infinitesimal is not widely used?

You ask "why Cauchy's definition of infinitesimal, along with his 'basic approach' was superseded?" The answer is that Cantor, Dedekind, Weierstrass and others developed a foundation for analysis to ...
• 43.3k
8 votes
Accepted

### Does absolute infinity invoke Cantor's Paradox?

First, it doesn't follow from Russell's Paradox that no set is a member of itself. It follows from the Axiom of Foundation, which is independent of the axioms necessary to prove Russell's Paradox. ...
• 23.9k

Only top scored, non community-wiki answers of a minimum length are eligible