Path: ...!weretis.net!feeder9.news.weretis.net!news.nk.ca!rocksolid2!i2pn2.org!.POSTED!not-for-mail
From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.logic
Subject: Re: Replacement of Cardinality
Date: Fri, 2 Aug 2024 12:12:08 -0400
Organization: i2pn2 (i2pn.org)
Message-ID: <5e690acdc28e7c22a458009b39e05884bf102602@i2pn2.org>
References: <hsRF8g6ZiIZRPFaWbZaL2jR1IiU@jntp>
 <cec0225a1e6ec21e1bca57b37fff99612e4505c4@i2pn2.org>
 <8G0IFYrPqHdBEH1pzbz9ifVRvd0@jntp>
 <11698e94cb8361b62f1686b64d6351a9720d4d3d@i2pn2.org>
 <nhZZyv1rDmL90pLuaDma-8md3qw@jntp>
 <1b259a91952c93a56ad1e0063a2d7440aed185f2@i2pn2.org>
 <rHIaB-dFODVqSY7-aRnf4ItTyG0@jntp>
 <20e0e340532aa10bcc86e51eb5d19d006acefb12@i2pn2.org>
 <el_h_RPLN1ZVr_KeaLK-R-0CPpY@jntp>
 <4d0ca88a910435926e85285b6a88fffe21ff9778@i2pn2.org>
 <kSBZ7VBqP15t7BtV3nwl-HtGQYU@jntp>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 8bit
Injection-Date: Fri, 2 Aug 2024 16:12:08 -0000 (UTC)
Injection-Info: i2pn2.org;
	logging-data="1215790"; mail-complaints-to="usenet@i2pn2.org";
	posting-account="diqKR1lalukngNWEqoq9/uFtbkm5U+w3w6FQ0yesrXg";
User-Agent: Mozilla Thunderbird
Content-Language: en-US
In-Reply-To: <kSBZ7VBqP15t7BtV3nwl-HtGQYU@jntp>
X-Spam-Checker-Version: SpamAssassin 4.0.0
Bytes: 4811
Lines: 110

On 8/2/24 11:09 AM, WM wrote:
> Le 02/08/2024 à 01:53, Richard Damon a écrit :
>> On 8/1/24 8:27 AM, WM wrote:
> 
>> And thus there is no "smallest" unit fraction, as for any eps, there 
>> are unit fractions smaller,
> 
> Your eps cannot be chosen small enough.

But you have the wrong definition of eps.

Between any two unit fractions, there is a finite non-zero eps that as 
smaller than their distance.

That does not say that one eps works for all unit fractions, that is 
just your invalid reversal of the conditions, that shows that your logic 
is broken.

> 
>>> That is the opinion of Peano and his disciples. It holds only for 
>>> potetial infinity, i.e., definable numbers.
>>
>> No, it holds for ALL his numbers.
> 
> Not for ℵo, i.e., for most it is wrong:
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo

ℵo is not a "Natural Number" or a number in Peano.

The fact that for any Natural Number, there are ℵo Natural Numbers above 
it is not a problem, unless you are using broken logic that think that 
ℵo obeys ALL the laws of finite mathematics.

> 
>>> What is the reason for the gap before omega? How large is it? Are 
>>> these questions a blasphemy?
>>
>> Because it is between two different sorts of number.
> 
> There is no gap above zero but e real continuum.

Because the real continuum is a single type of number.

>>
>> There is a gap between 1 and 2, but that doesn't bother you.
> 
> All gaps of size 1 do not bother me..

Good, Then when talking about omega, gaps between numbers isn't a 
problem either, we get the sets of 0*Omega + n, as the Natural Numbers, 
then 1*Omega + n as the first set of transfinite numbers, nd those 
having a gap of Omega shouldn't be a problem.

>>> It is the definition of definable numbers. Study the accumulation 
>>> point. Define (separate by an eps from 0) all unit fractions. Fail.
>>
>> So, which Unit fraction doesn't have an eps that seperates it from 0?
> 
> There are infinitely many by the definition of accumulation point. You 
> cannot find them. Therefore they are dark.

Nope, we can find any one of them we want.

Again, you are changing the conditional incorrectly because you logic 
can't handle unbounded values.

>>
>> You just get your order of conditions reversed.
> 
> I get it the only corect way. Every eps that you can chose belongs to a 
> set of chosen eps. This set has a minimum - at every time. It is finite. 
> Quantifiers therefore can be reversed.

And that is your problem, the set of eps doesn't HAVE a minimum, because 
it is unbounded.

You logic just can't handle unbounded sets, and thus can't actually have 
any infinity,

>>
>> For all 1/n, there is a eps that is smaller than it (like 1/(n+1) )
> 
> For all 1/n that you can define.

For *ALL* 1/n, PERIOD, since I can define any of them.

>>
>> And for every eps, there is a unit fraction smaller than it
> 
> There are infinitely many, namely almost all.

No, FOR ALL.

Name the one that isn't

> 
>> So we have an unlimited number of Unit fractions, and no smallest one.
> 
> But you have a limited number of eps.
> 

Nope, why do you say I have a limited number of eps?

That is just you proving your stupidity, and that you are stuck in a 
finite and bounded logic system trying to handle something that is 
actually unbounded.

> Regards, WM
> 
>