Why Marx Can Never Pay off his Debts: A Version of Yablo’s Paradox (1993)

Marx needs more than $99 to pay off his debts, but Engels will only give him exactly $99. For any positive amount X Marx can muster over and above Engels’s $99 hand-out, under capitalism, there is a corresponding petit bourgeoisie “thug” to whom Marx must pay all but a fraction of X.

What Marx must pay each petty bourgeoisie thug Ti for every additional unit X over and above the $99 hand-out from Engels is determined as follows:

1. If Marx earns amount X, then as (X + $99) > $99 he must pay all but X/2 to T1.

2. If (X /2 + $99) > $99 then he must pay all but X/4 to T2.

3. If (X/4 + $99) > $99, then he must pay all but X/8 to T3.

And so on.

If Marx doesn’t earn anything apart from Engels’s $99 hand-out, or if his earnings along with the same $99 sum to <$99 then he has to pay nothing to any petty bourgeoisie thugs.

Given this state of affairs it turns out that Marx will never be able to clear his debts; this is unsurprising. What is surprising, however, is that Marx will always remain indebted without ever paying petty bourgeoisie thugs anything—as his income will never exceed the $99 hand-out from Engels.

PROOF: Suppose Marx made an amount X = 1 cent, so his net worth along with Engels’s hand-out amounted to $99.01. Then since that value was >$99 by $0.01 Marx would have to pay T1 all but $0.005; leaving him with $99.005. But as that too would be >$99 Marx would have to then pay T2 all but $0.0025 from that 1 cent. And, as he’d now have $99.0025 which is > $99, he’d have to pay T3 all but $0.00125 from the same 1 cent leaving him with $99.00125. Since $99.00125 > $99, Marx would further have to pay T4 $0.000625, and since $99. 000625 > $99 he’d also have to pay all but $ 0.0003125 to T5, leaving him with $99.0003125. Similarly, Marx would always have to pay all but the 1/knth part of X to thugs Tk for arbitrarily large k values, as the fraction left over would always continue to approach 0, without becoming equal to 0, as n approached infinity. So, contrary to our initial supposition, Marx couldn’t ever earn anything more than Engels’s $99 hand-out and so would stay forever in debt, but since his earning would never be >$99 he would also pay nothing to any petty bourgeoisie thugs.

Inspired by Roy T. Cook’s brief but excellent article: http://blog.oup.com/2015/07/yablo-bernardete-paradox/

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