How to invoke constants badly

It came to my mind what's perhaps the oldest example of this kind of mistake, so I add an answer to my own question: in 1821 Cauchy 'proved' that convergent sums of continuous functions are continuous, and later on Abel found counterexamples (see [1] for historical details). Of course Cauchy implicitly assumed uniform convergence, which means that he treated his $\delta$ as "more constant" than it was...

[1]: Sørensen, H. K. (2005). Exceptions and counterexamples: Understanding Abel's comment on Cauchy's Theorem. Historia Mathematica, 32(4), 453-480.

A good example (of a somewhat different kind though) was given by Adian in the introduction to his book "The Burnside problem and identities in groups" (1975, English edition 1979), where he refutes the proof of the main result from his rival's book by stating that

``the conditions $$ \begin{aligned} &u_4 = u_1 +r_{25} \; \text{(p.145, line 10 from below)} \\ &r_{25} \ge u_{37} +54/e \; \text{(p.283, line 4 from below)} \\ &u_{37} >14\alpha +214/e, \; \text{where}\; \alpha=\varepsilon_{30}+u_{13} +6u_4 \; \text{(p.221, lines 11 and 12 from below)} \end{aligned} $$ give an obvious contradiction $u_4>r_{25}>u_{37}>u_4$.''

Edit: The original answer below refers to Nelson's attempt from 2011. Upon a cursory look at the afterword by Sam Buss and Terence Tao to Nelson's paper placed in arxiv in 2015 (after his death), it seems he later attempted to address the error referred to in the original answer below; it would be interesting to know what the experts think on how successful his efforts were or potentially can be.

Original Answer: Edward Nelson's recent project on finding inconsistency of arithmetic (which was the subject of a MathOverflow Question) might be pertinent. The error, discovered by Terence Tao, seems to be the dependence of a constant on the underlying theory that Nelson did not account for.