Consider the real line $\mathbb R$ and $C^1_0$ , the ring of germs of continuously differentiable functions at zero.
Now take the ideal $M$ of germs vanishing at zero. The Zariski cotangent space $M/M^2$ has dimension the continuum (because the classes of
$x^{1+\epsilon}$
are linearly independent in the quotient
for $0<\epsilon <1$ ).
Hence the Zariski tangent space of the real line, i.e. the dual of $M/M^2$, has dimension $2^{continuum}$. Some geometers might think this is a bit large for the real line.
This result is essentially exercise 13 of Chapter 3 of Spivak's Differential Geometry, Volume I.
In a more recent paper of Friedman
Friedman, Harvey M., Concept calculus: much better than, Heller, Michael (ed.) et al., Infinity. New research frontiers. Based on the conference on new frontiers in research on infinity, San Marino, August 18–20, 2006. Cambridge: Cambridge University Press (ISBN 978-1-107-00387-3/hbk). 130-164 (2011). ZBL1269.03008,
the author defines a mathematically precise system MBT (much better than) and proves it and ZF have mutual interpretability. This establishes that if either is consistent they both are. These axioms have some of the flavor of IP and PP, but of course these axioms are not implied by IP and PP.
At the end of the paper Friedman claims a to be published result. STAR is defined as:
There exists a star. I.e., something which is better
than something, and much better than everything it is better than.
We have shown that MBT + STAR can be interpreted in some large cardinals compatible with V = L, and some large cardinals compatible with V = L are interpretable in MBT + STAR."
For PP and IP to be true, in a sense that can prove mathematics, they need to be stated precisely like the axioms in MBT. That formulation is much more complex than PP and IP as it must be to interpret ZF.
It is important to keep in mind that consistency does not imply truth. The statement that a formal system is consistent is equivalent to a statement of the form $\forall_{n\in\omega} r(n)$ where $r$ is a recursive relationship. This is equivalent to the halting problem for a particular Turing machine.
The following quote from Friedman is, I suspect, a big part of his and others interest in this work:
STARTLING OBSERVATION. Any two natural theories S,T, known to interpret PA, are known (with small numbers of exceptions) to have: S is interpretable in T or T is interpretable in S. The exceptions are believed to also have comparability.
It is an interesting and even startling observation, but it is worth keeping in mind that that rigorous theories are, among other things, recursive processes for enumerating theorems. To say that one theory is interpretable in another is to say a subset of one processes outputs are, in a specific well defined sense, isomorphic to the outputs of the process defined from the theory being interpreted. Whatever other significance it may have, this is a statement about unbounded recursive processes.
My personal view (see what is Mathematics About?) is that the only mathematics that can be interpreted as a properties of recursive processes is objectively true or false. This is based on the old idea that infinite is a potential that can never be realized. In this view Cantor's proof that the reals are not countable is an incompleteness theorem. The cardinal hierarchy is a hierarchy of the ways the real numbers provably definable in a formal system can always be expanded. Because of the Lowheheim Skolem theorem, we know such an interpretation exists. Interpretations that assume the absolutely uncountable are inevitably ambiguous at least as far as they can be expressed formally in the always finite universe that we seem to inhabit.
Best Answer
Lie's third theorem - that every finite-dimensional Lie algebra (over the reals) is the tangent space of some Lie group - is quite difficult to establish without first establishing Ado's theorem that every finite-dimensional Lie algebra embeds into the Lie algebra of a general linear group. But once Ado's theorem is in place, the claim is quite easy: the Lie group is the space of curves in the general linear group that are tangent to the embedded linear algebra, up to smooth deformation.
(Without Ado's theorem, it is relatively straightforward to make the Lie algebra the tangent space of a local Lie group via the Baker-Campbell-Hausdorff formula, but establishing the global associative law necessary to extend this local Lie group to a global one is highly non-trivial. The main contribution of Ado's theorem, then, is allowing one to exploit the global associativity of the general linear group, which is a triviality in this concrete setting, but not in the setting of the abstract local group given by BCH.)