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(Work on the Lambda-CDM metric)

The FLRW metric with two spatial dimensions suppressed is

Ignoring the effects of radiation in the early universe and assuming k = 0 and w = −1, the Lambda-CDM scale factor is

Putting (for reasons that will emerge later)

and

,

the Lambda-CDM scale factor may be rewritten as

Formally expanding the binomial and simplifying gives

Ratio of successive terms = which tends to as n tends to infinity.


Best Current Numerical Values

The WMAP five-year report gives

(Mp = megaparsec, Ga = gigayear).

These give

and


The path of the light ray satisfies .



________________________________________________________________________________________

An Apparent Contradiction (Unfortuanately this Word file contains special characters that won't print here. )


The following refers to "The Emperor's New Mind", OUP 1989(99), chapter 4, section 3 (Gödel’s Theorem). On page 140, there is a statement, derived on the previous pages: ~x[x proves Pk(k)] = Pk(k) With the simple substitution Pk(k) = S made to simplify the analysis and since k doesn’t feature explicitly in what follows, this is my assumption 00. There are two other assumptions: 01 and 02. From these three assumptions a contradiction emerges on lines 09 and 17. The question is: where and why does the contradiction arise? The following uses a modified version of the scheme used in the OU course “Number Theory & Mathematical Logic”.

Line Statement Derivation/Comments Assumptions used

00 ~x [x proves S] = S Assumption 00 01 [x proves S] = [x . [x  S]] Assumption 01 02 A x[A = x] Assumption 02 03 ~x[x . [x  S]] = S Subs (0100) 00,01 04 ~S Assumption 04 05 x[x . [x  S]] Subs/Taut (03,04) 00, 01, 04 06 y . [y  S] Quant’r Removal (05) 00, 01, 04 07 S Taut (06) 00, 01, 04 08 ~S  S Proof (04,07) 00, 01 09 S Taut (08) 00, 01 10 [~x [x proves S] = S]  S Proof (00,09) 01 11 x[[~x [x proves S] = S] = x] Quant’r Removal (02) 02 12 [~x [x proves S] = S] = y Quant’r Removal (11) 02 13 y Subs (1200) 00, 02 14 y  S Subs (1210) 01, 02 15 y . [y  S] Taut (13,14) 00, 01, 02 16 x[x . [x  S]] Quant’r Insertion (15) 00, 01, 02 17 ~S Taut (16,03) 00, 01, 02


HTML VERSION:

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An Apparent Contradiction<o:p></o:p>

<o:p> </o:p>

<o:p> </o:p>

The following refers to "The Emperor's New Mind", OUP 1989(99), chapter 4, section 3 (Gödel’s Theorem). <o:p></o:p>

<o:p> </o:p>

On page 140, there is a statement, derived on the previous pages: <o:p></o:p>

<o:p> </o:p>

~$x[Õx proves Pk(k)] = Pk(k)<o:p></o:p>

<o:p> </o:p>

With the simple substitution Pk(k) = S made to simplify the analysis and since k doesn’t feature explicitly in what follows, this is my assumption 00.  There are two other assumptions: 01 and 02. From these three assumptions a contradiction emerges on lines 09 and 17.  The question is: where and why does the contradiction arise?   <o:p></o:p>

<o:p> </o:p>

The following uses a modified version of the scheme used in the OU course “Number Theory & Mathematical Logic”.  <o:p></o:p>

<o:p> </o:p>

<o:p> </o:p>

Line<o:p></o:p>

Statement<o:p></o:p>

Derivation/Comments<o:p></o:p>

Assumptions used<o:p></o:p>

<o:p> </o:p>

<o:p> </o:p>

<o:p> </o:p>

<o:p> </o:p>

00<o:p></o:p>

~$x [Õx proves S] = S<o:p></o:p>

Assumption<o:p></o:p>

00<o:p></o:p>

01<o:p></o:p>

[Õx proves S] = [Õx . [Õx Þ S]]<o:p></o:p>

Assumption<o:p></o:p>

01<o:p></o:p>

02<o:p></o:p>

"A $x[A = Õx]<o:p></o:p>

Assumption<o:p></o:p>

02<o:p></o:p>

03<o:p></o:p>

~$x[Õx . [Õx Þ S]] = S<o:p></o:p>

Subs (01®00)<o:p></o:p>

00,01<o:p></o:p>

04<o:p></o:p>

~S<o:p></o:p>

Assumption<o:p></o:p>

04<o:p></o:p>

05<o:p></o:p>

$x[Õx . [Õx Þ S]]<o:p></o:p>

Subs/Taut (03,04)<o:p></o:p>

00, 01, 04<o:p></o:p>

06<o:p></o:p>

Õy . [Õy Þ S]<o:p></o:p>

Quant’r Removal (05)<o:p></o:p>

00, 01, 04<o:p></o:p>

07<o:p></o:p>

S<o:p></o:p>

Taut (06)<o:p></o:p>

00, 01, 04<o:p></o:p>

08<o:p></o:p>

~S Þ S<o:p></o:p>

Proof (04,07)<o:p></o:p>

00, 01<o:p></o:p>

09<o:p></o:p>

S<o:p></o:p>

Taut (08)<o:p></o:p>

00, 01<o:p></o:p>

10<o:p></o:p>

[~$x [Õx proves S] = S] Þ S<o:p></o:p>

Proof (00,09)<o:p></o:p>

01<o:p></o:p>

11<o:p></o:p>

$x[[~$x [Õx proves S] = S] = Õx]<o:p></o:p>

Quant’r Removal (02)<o:p></o:p>

02<o:p></o:p>

12<o:p></o:p>

[~$x [Õx proves S] = S] = Õy<o:p></o:p>

Quant’r Removal (11)<o:p></o:p>

02<o:p></o:p>

13<o:p></o:p>

Õy<o:p></o:p>

Subs (12®00)<o:p></o:p>

00, 02<o:p></o:p>

14<o:p></o:p>

Õy Þ S<o:p></o:p>

Subs (12®10)<o:p></o:p>

01, 02<o:p></o:p>

15<o:p></o:p>

Õy . [Õy Þ S]<o:p></o:p>

Taut (13,14)<o:p></o:p>

00, 01, 02<o:p></o:p>

16<o:p></o:p>

$x[Õx . [Õx Þ S]]<o:p></o:p>

Quant’r Insertion (15)<o:p></o:p>

00, 01, 02<o:p></o:p>

17<o:p></o:p>

~S<o:p></o:p>

Taut (16,03)<o:p></o:p>

00, 01, 02<o:p></o:p>

<o:p> </o:p>

<o:p> </o:p>

</body>

</html> --Dendropithecus (talk) 23:14, 21 May 2010 (UTC)