CAPSTONE: Non-standard analysis

Section 25.7 CAPSTONE: Non-standard analysis

The hyperreal numbers, denoted $\hypext\bb R\text{,}$ are a number system where the real numbers have been extended by infinite and infinitesimal elements.
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The whole idea was to make Newton and (especially?) Leibniz’s ideas of infinitesimals rigorous.
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It’s actually pretty easy to algebraically extend the real numbers by adding an infinitesimal element. The difficult comes in extending things like transcendental functions to work with these new numbers.
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Developed by Abraham Robinson in the 1960s
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Extension principle:
- The real numbers form a subset of the hyperreal numbers, and the order relation $x<y$ for the real numbers is a subset of the order relation for the hyperreal numbers.
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- There is a hyperreal number that is greater than zero but less than every positive real number.
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- For every real function $f$ of one or more variables, there exists a corresponding hyperreal function $\hypext\!f$ of the same number of variables, called the natural extension of $f\text{.}$ For any $x\in\bb R\text{,}$ we have $\hypext\!f(x)=f(x)\text{.}$
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Transfer principle:
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Essentially, algebraic formulas are true in the real numbers if and only if they’re true in the hyperreal numbers.
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Basic laws of arithmetic: if $h$ is infinitesimal, $f$ is limited (finite but not infinitesimal), and $H$ is infinite:
- $h^{2}$ is infinitesimal
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- $H^{2}$ is infinite
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- $1/h$ is infinite
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- $1/H$ is infinitesimal
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- $h\cdot f$ is infinitesimal
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- $H\cdot f$ is infinite
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- $H\cdot h$ may be infinite, infinitesimal, or limited
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Define the following symbols:
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\begin{align*} a \amp \ll b \amp \text{$a$ is infinitesimal with respect to $b$}\\ a \amp \approx b \amp \text{$a-b$ is infinitesimal} \end{align*}
Then we have the following:
- $\displaystyle h\ll f\ll H$
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- $\displaystyle f+h\approx f$
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The standard part of a hyperreal number $h\text{,}$ denoted $\st h\text{,}$ is the unique real number $r$ such that $r\approx h\text{.}$ This function is external.
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Continuity: A function $f$ is continuous at $x\in\bb R$ if $\hypext\!f(x+h)\approx\hypext\!f(x)$ whenever $h$ is infinitesimal. (This is called microcontinuity.)
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Uniform continuity: A function $f$ is uniformly continuous if whenever $x\in\hypext\bb R$ and $h$ is infinitesimal, $\hypext\!f(x+h)\approx \hypext\!f(x)\text{.}$
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Derivative: If $h$ is infinitesimal, then

\begin{equation*} f'(x)=\st\frac{\hypext\!f(x+h)-\hypext\!f(x)}{h}\text{.} \end{equation*}

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Extended subsets: $\hypext\bb N,\hypext\bb Z,\hypext\bb Q$ (hypernatural, hyperinteger, hyperrational)
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Integral:

\begin{equation*} \int_{a}^{b} f(x)\dd{x}=\st\sum_{k=1}^{N}\hypext\!f(x_{k})\Delta x_{k} \end{equation*}

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Construction:
- Start with the collection of all sequences in $\bb R$
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- Define addition and multiplication pointwise
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- Constant sequences are associated with real numbers
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- Comparisons are based on individual comparisons on a “large” number of terms:
  - For any set $S\text{,}$ either $S$ is large or its complement is large.
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  - No finite sets are large / all cofinite sets are large.
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  - Large sets are closed under intersections.
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  - Large sets are upward closed; that is, if $A$ is large and $A\subseteq B\text{,}$ then $B$ is large.
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  A specification of these “large” sets is called a nonprincipal ultrafilter on $\bb N\text{.}$ A nonprincipal ultrafilter exists if you assume Zorn’s lemma, and it’s unique up to isomorphism if you assume the Continuum Hypothesis. (May need to include countability somewhere in the RA section as a result if I want to say this.)
  
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- From here, $\omega=(1,2,3,\cdots)$ is infinite, and $1/\omega=(1,\tfrac12,\tfrac13,\cdots)$ is infinitesimal.
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