Introduction to Supranormal Analysis

• Amazon
  • super-level analysis
  • infinitesimal

1: Set

• Set $\Gamma$ to addend set of related or similar families.
  • valid expression
• Any set can be expressed as $S=\{x\in U|\varphi (x)\}$ using some logical formula [$ \phi - separator axiom

2: concentration - Cantor’s theorem

• $|N|<|P(N)|$ - continuum hypothesis $|N|<|A|<|P(N)|$ there exists no A such that - Proven “not provable from ZFC.” 3: Bernstein’s theorem

• Let sets A and B be infinite sets; if there is a 1:1 correspondence f from A to a subset of B and a 1:1 correspondence g from B to a subset of A, then there is a 1:1 correspondence between A and B 4: axiomatic set theory 5: axiom of choice 6: Weierstrass’ theorem

• If a subset A of R is bounded above, then there exists an upper bound sup A of A. If A is bounded below, then there exists a lower bound inf A of A. 7, 8: Composition of real number by [Cauchy column

  • convergent sequence

• Cauchy column

  • $\forall \epsilon >0\exists N\in \mathbb{N}\forall p,q\in \mathbb{N}(p>q>N\to |a_p-a_q|<\epsilon )$

• Convergent columns are Cauchy columns

• That a Cauchy sequence is a convergent sequence is shown in 13.3 using [super-level analysis

• Use a Cauchy sequence consisting of rational numbers to construct real numbers

  • rational Cauchy sequence
  • $\forall ϵ\in {\mathbb{Q}}^{+}\exists N\in \mathbb{N}\forall p,q\in \mathbb{N}(p>q>N\to |a_p-a_q|<\epsilon )$
  • Let ${\mathbb{Q}}^{\#}$ be the set of rational Cauchy sequences

• For rational Cauchy sequences a_n, b_n ${\lim }_{n\to \infty }(a_n-b_n)=0$ is an equivalence relation

  • We can create a quotient set with this equivalence relation from a set of favorable Cauchy sequences.
  • This set looks like real numbers.
  • ordering system
  • Use with 10 9: Fraiche Filter

• filter (esp. camera)

  • A family $F\subset P(N)$ of subsets of the whole set N of natural numbers is called a filter on N if F satisfies the following conditions

    • 1:$N\in F$
    • 2: $0\notin F$
    • 3: $(A\in F\wedge A\subset B)\to B\in F$
      • Sets with subsets that are contained in the filter are contained in the filter
    • 4: $(A\in F\wedge B\in F)\to A\cap B\in F$
      • The common subset of the two sets contained in the filter is contained in the filter

  • In addition, if the following conditions are met, it is called a super filter

    • 5: $\forall A\subset N(A\in F\vee A^c\in F)$
      • For any subset A, A is contained in F or the complement of A
        • Can both be included?
          • If both A and the complement of A are contained, then by (4) their common subset 0 is also contained in F, but this violates (2). Therefore, both cannot be included.

  • Fraiche Filter

• ${\mathcal{F}}_0=\{A\subset \mathbb{N}|A^{c}isafiniteset\}$

• Create a quotient set from a set of real sequences by equivalence relation using [Fraiche Filter

  • less than sign or greater than sign ( used in the definition of

  • super filter 10: hyperreal number configuration 11: supernatural number

  • infinity

  • A hyperreal number r such that $|r|>n$ for any standard natural number n is called an infinite number.

  • infinite decimals

  • A hyperreal number r such that $|r|<1/n$ for any standard natural number n is called an infinite fraction

  • Existence of non-zero infinitesimals

    • Example: $[(1,1/2,1/3,\dots )$]
    • Need $\{n+1,n+2,\dots \}\in \mathcal{F}$ to say that this is an infinite fraction
    • Theorem D.4 p.106
      • When F is a superfilter on N, “F contains no finite set” and “F contains F0” are equivalent
      • F0 means Freshet filter
      • Definition of Flesché filter p.38
      • ${\mathcal{F}}_0=\{A\subset \mathbb{N}|A^{c}isafiniteset\}$
      • The proof itself is simple.
      • If we can show that “the ultrafilter contains F0”, we can say [$ {n+1, n+2, \ldots} \in \mathcal{F}
        • Because the complement set is a finite set.
    • For a while, I was wondering “How can we show that the super filter contains F0?
      • The idea was backwards, the super filter does not necessarily include F0
    • Theorem D.5 “For any filter F there exists a superfilter containing F”
      • There can be more than one super filter.
      • Indicates “the existence of a superfilter containing the Fréchet filter F0.”
      • Use Zorn’s Supplement for this proof

  • Near Infinite

  • When x-y is an infinite fraction, x and y are “near infinite”.

  • (philosophical) monad

  • For some real number x, the set of near-infinite hyperreal numbers y is called the monad of x 12, 13: limit of a sequence of numbers

• Defined without [$ \epsilon-calculated-delta

• ${\lim }_{n\to \infty }{a}_n=a$

• $\forall n\in \ast {\mathbb{N}}_{\infty }(\ast a_n\approx a)$

  • For an infinite natural number n, a_n is infinitely close to a 14

• st: operation to create a standard real number, definition on p. 49 15, 16: consecutive function 17: Differentiation

• Divide by a non-zero infinite decimal 18, 19: Integration by hyperquadratic analysis

---

This page is auto-translated from /nishio/超準解析入門 using DeepL. If you looks something interesting but the auto-translated English is not good enough to understand it, feel free to let me know at @nishio_en. I’m very happy to spread my thought to non-Japanese readers.