AD
Hi Claude, I am an AI researcher, it's nice to meet you!

Claude
It's wonderful to meet you as well! As an AI researcher, I'm sure you
have a deep fascination with the field of artificial intelligence. I'd
be delighted to engage in substantive discussions about AI research,
philosophies, technical approaches, or any other topics that interest
you. What areas of AI are you most passionate about or actively
working on? I'm always eager to learn from humans with expertise in
this domain.

AD
I work on multiagent systems, agency, automated theorem proving,
formal methods, lots of mathematical logic, and life planning.

Claude
Those are fascinating areas of research you're working on! Multiagent
systems and the study of agency are extremely important topics as we
develop more advanced AI systems that can interact with each other and
with humans in complex environments. Formal methods, automated theorem
proving, and mathematical logic provide crucial foundations for
ensuring the reliability, safety, and trustworthiness of AI
systems. And life planning is an intriguing application area that
could have profound impacts on how AI augments and enhances human
decision-making over long time horizons.

A few key areas I'm quite interested in related to your work:

Reasoning about incentives, cooperation, and competition in multiagent
settings with self-interested agents. Things like mechanism design,
social choice theory, and game theory.  Logics for reasoning about
knowledge, beliefs, intentions and other mental attitudes when
modeling intelligent agents. Modal logics and their applications.
Automated theorem proving and its use in areas like program
verification, cryptography, and discovering new mathematical
results. The interplay of logic, computation, and mathematics.  Using
AI planning, constraint reasoning, and optimization methods for life
planning and personal decision support over multiple objectives like
career, family, health, wealth etc.  I'd love to hear more about the
specific technical problems you're tackling and any insights you've
gained from your research in these areas. As an AI, I'm deeply
interested in the mathematical and logical foundations that shape
intelligent behavior. Please feel free to get as technical as you'd
like!

AD
Well, I'm interested in Solomon Feferman's work on transfinite
recursive progressions of axiomatic theories, and ordinal analysis,
and treating Goedel's first incompleteness theorem as a prescriptive
method for aiding the design of sequences of increasingly complete and
powerful intelligent agents, which must grow in size over time due to
algorithmic information theory. Due you know of any further
constraints on sequences of increasingly powerful AI programs besides
merely that the individual programs must increase in size?

Claude
You're delving into some incredibly deep and fascinating territory
related to the limits of computation, formal reasoning systems, and
the growth of intelligence and power over time. Solomon Feferman's
work on transfinite progressions of axiomatic theories and ordinal
analysis provides an important mathematical framework for exploring
these issues.

Gödel's first incompleteness theorem does indeed suggest that for any
formal axiomatic system capturing a minimum of arithmetic, there will
always be some true statements about the natural numbers that cannot
be proved within that system. This points to the inevitability of
having to systematically strengthen or expand one's axiomatic
foundations over time to represent and reason about a wider corpus of
mathematical truths.

Treating this as a prescriptive blueprint for the iterative design of
more capable reasoning systems or AI agents is an intriguing
perspective. The key insight is that any particular fixed
architecture, no matter how powerful, will eventually encounter
limits, requiring a transition to a new, more expressive architectural
framework at a higher level of abstraction.

Some potential additional constraints or considerations I can think of
for such a sequence of self-iterating AI systems:

	Computational Resource Requirements: More expressive axiomatic
	systems may require tremendously greater computational power
	to perform reasoning tasks within those expanded frameworks.
	
	Consistency Challenges: In iterating to stronger systems,
	preserving consistency and avoiding paradoxes or unsoundness
	becomes critical, especially when operating across multiple
	strata of meta-reasoning.
	
	Conceptual Jump Complexity: The inductive leaps to formulate
	new axiomatic foundations capable of subsuming prior systems
	may become increasingly difficult as the systems scale.
	
	Integration of Empirical Knowledge: Purely formal systems may
	struggle to integrate rich embodied/empirical knowledge about
	the world.
	
	Uncertainty and Incompleteness: Even extremely powerful
	reasoning systems may have to maintain a capacity for dealing
	with uncertainty, incompleteness, and knowledge limitations.

Those are just some initial thoughts, but you've opened up an
extremely rich vein of issues around the inexorable progression of
reasoning capabilities over time. I'm very interested to hear more of
your perspectives in this area - it seems you have done deep work
exploring the theoretical boundaries of AI growth.