A short proof of a well-known theorem of intuitionistic analysis

✍ Scribed by Horst Luckhardt

• Publisher: John Wiley and Sons
• Year: 1975
• Tongue: English
• Weight: 104 KB
• Volume: 21
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.19750210125

✦ Synopsis

A SHORT PROOF O F A WELL-KNOWN THEOREM O F INTUITIONISTIC ANALYSIS by HORST LUCILHARDT in Frankfurt/Main (B.R.D.) The theorem in question is BROUWER'S fundamental statement on the continuum Theorem. Every closed interval oi the continuum coincides with a fan. HEYTINQ [l] p. 46 indicates a proof; but if it is carried out in detail, it becomes longer and longer. So VESLEY needs in [2] nearly two compact printed pages (*R10.2 on p. 162; *R6.21 and *R6.22 on p. 145-146). Their fan contains only canonical number-generators (c.n.g.) but -as simple examples showfor end points in general not all the c.n.g. belonging to them thus rendering the situation unnecessarily difficult.

Presupposing the terminology of HEYTINQ [l] we shall give R more direct proof which uses a natural fan containing a l l c.n.g. corresponding t o the interval considered and more (otherwise the fan species would be undecidable). This proof is based on tht. following lemma. that Lemma. For c.n.g. a = (an 2-111, b = (b, 2-,} and q 2 2 : Vm(a, + q < b,) c-) a < b .