Re: Relations as primitive
From: Bob Badour <bbadour_at_pei.sympatico.ca>
Date: Fri, 11 Jan 2008 19:49:50 -0400
Message-ID: <478800a3$0$19873$9a566e8b_at_news.aliant.net>
>>Marshall wrote:
>>
>>>On Jan 11, 12:54 am, David BL <davi..._at_iinet.net.au> wrote:
>>
>>>>[...]
>>
>>>The mathematical idea of functions is the set of ordered
>>>pairs, where the left item in the pair is a member of the
>>>domain and the right item is a member of the range.
>>>The domain can then be considered also to be ordered
>>>pairs, so that + then looks like
>>
>>> {((1,1),2), ((2,2),4) ... }
>>
>>>And so on. This is a consequence of the foundational
>>>approach in which sets are the basic building blocks.
>>>(And ordered pairs are defined in terms of sets, via
>>>Kuratowski or whatever.)
>>
>>>However there is another way to go, which I first
>>>encountered in ch. 4 of TTM, and which I don't see
>>>discussed much or at all in mathematical circles,
>>>and which I find immensely appealing. The approach
>>>is to work directly with relations. So + looks like
>>
>>> {(x=1, y=1, z=2), (x=2, y=2, z=4) ...}
>>
>>>In point of fact, there is almost nothing of significance
>>>to this idea; it's equivalent to the usual formulation.
>>>But I nonetheless find the idea appealing of thinking
>>>about the foundations of mathematics in terms of
>>>taking relations as primitive rather than sets.
>>
>>Relations are sets.
Date: Fri, 11 Jan 2008 19:49:50 -0400
Message-ID: <478800a3$0$19873$9a566e8b_at_news.aliant.net>
Marshall wrote:
> On Jan 11, 12:21 pm, Bob Badour <bbad..._at_pei.sympatico.ca> wrote: >
>>Marshall wrote:
>>
>>>On Jan 11, 12:54 am, David BL <davi..._at_iinet.net.au> wrote:
>>
>>>>[...]
>>
>>>The mathematical idea of functions is the set of ordered
>>>pairs, where the left item in the pair is a member of the
>>>domain and the right item is a member of the range.
>>>The domain can then be considered also to be ordered
>>>pairs, so that + then looks like
>>
>>> {((1,1),2), ((2,2),4) ... }
>>
>>>And so on. This is a consequence of the foundational
>>>approach in which sets are the basic building blocks.
>>>(And ordered pairs are defined in terms of sets, via
>>>Kuratowski or whatever.)
>>
>>>However there is another way to go, which I first
>>>encountered in ch. 4 of TTM, and which I don't see
>>>discussed much or at all in mathematical circles,
>>>and which I find immensely appealing. The approach
>>>is to work directly with relations. So + looks like
>>
>>> {(x=1, y=1, z=2), (x=2, y=2, z=4) ...}
>>
>>>In point of fact, there is almost nothing of significance
>>>to this idea; it's equivalent to the usual formulation.
>>>But I nonetheless find the idea appealing of thinking
>>>about the foundations of mathematics in terms of
>>>taking relations as primitive rather than sets.
>>
>>Relations are sets.
> > Sure. > > Relations are a specific kind of set. The usual foundational > approach is to take (general) sets as primitive. However > we could instead take a more specific kind of set as > primitive, namely relations. > > Marshall
I am not sure what we hope to accomplish by doing so. Goedel awaits at the end of the line regardless.
Certainly, I take comfort in the power of the relational formalism. I try not to get too hung up on the philosophy of mathematics stuff. Received on Sat Jan 12 2008 - 00:49:50 CET