Re: Is 0.123... a member of the set {0, 0.1, 0.12, 0.123, ...}?

From: Will Twentyman (wtwentyman_at_read.my.sig)
Date: 02/01/05 

Date: Tue, 01 Feb 2005 17:48:37 -0500

Don Whitehurst wrote:

> Is 0.123... a member of the set {0, 0.1, 0.12, 0.123, ...}?
>
> Consider the following series of paired sets Ai and Bi.
>
> For n = 1,
> Set A1 = { 0, 0.1 }
> Set B1 = { 0.1 }
> Set B1 is a subset of Set A1.
>
> For n = 2,
> Set A2 = { 0, 0.1, 0.12 }
> Set B2 = { 0.12 }
> Set B2 is a subset of Set A2.
>
> For n = 3,
> Set A3 = { 0, 0.1, 0.12, 0.123 }
> Set B3 = { 0.123 }
> Set B3 is a subset of Set A3.
>
> Consider what happens when set A becomes an infinite set;
> Set Ainf = { 0, 0.1, 0.12, 0.123, ... }
> Set Binf = { 0.123... }
> Set Binf is a subset of Set Ainf.
>
> 0.1234567891011121314... is a member of the infinite set { 0, 0.1,
> 0.12, 0.123, ...}.

You are making your argument based on induction. However, induction
does not warrant the logical leap you have just made.

>
> Now consider the following series of paired sets Ci and Di.
>
> For n = 1,
> Set C1 = { 0, 0.3 }
> Set D1 = { 0.3 }
> Set D1 is a subset of Set C1
>
> For n = 2
> Set C2 = { 0, 0.3, 0.31 }
> Set D2 = { 0.31 }
> Set D2 is a subset of Set C2
>
> For n = 3
> Set C3 = { 0, 0.3, 0.31, 0.314 }
> Set D3 = { 0.314 }
> Set D3 is a subset of Set D3
>
> Consider what happens when set C becomes an infinite set:
> Set Cinf = { 0, 0.3, 0.31, 0.314, ... }
> Set Dinf = { 0.314... }
> Set Dinf is a subset of Set Cinf.
>
> Pi/10 = 0.314... is a member of the infinite set { 0, 0.3, 0.31, 0.314,
> ...}.

You are making your argument based on induction. However, induction
does not warrant the logical leap you have just made.

> Now consider the following series of paired sets Ei and Fi.
>
> For n = 1,
> Set E1 = { 0, 0.3 }
> Set F1 = { 0.3 }
> Set F1 is a subset of Set E1
>
> For n = 2
> Set E2 = { 0, 0.3, 0.33 }
> Set F2 = { 0.33 }
> Set F2 is a subset of Set E2
>
> For n = 3
> Set E3 = { 0, 0.3, 0.33, 0.333 }
> Set F3 = { 0.333 }
> Set F3 is a subset of Set E3
>
> Consider what happens when set E becomes an infinite set:
> Set Einf = { 0, 0.3, 0.33, 0.333, ... }
> Set Finf = { 0.333... }
> Set Finf is a subset of Set Einf.
>
> 1/3 = 0.333... is a member of the infinite set { 0, 0.3, 0.33, 0.333,
> ...}.
>
>
> Similar arguments can be used to show that any repeating decimalic
> rational or irrational number expressed in decimalic form whose value
> lies between 0 and 1 is a subset of a properly constructed set
> containing a series that converges to that specific (any) number.
>
> Obviously, this must either be wrong or the naturals can map to the
> reals.
>
> Where did I go wrong?

The correct conclusion is that for all n in N, Fn is a subset of En. 

-- 
Will Twentyman
email: wtwentyman at copper dot net

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